ELEMENTARY

FOREST

SAMPLING

/

-1 ,' '-vi I

Agriculture Handbook No. 232

U.S. Department of Agriculture • Forest Service
ELEMENTARY

FOREST

SAMPLING

FRANK FREESE

Southern Forest Experiment Station, Forest Service

Agriculture Handbook No. 232 December 1962

U.S. Department of Agriculture • Forest Service

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington 25, D.C. - Price 50 cents
ACKNOWLEDGMENTS

I should like to express my appreciation to Professor George W.

Snedecor of the Iowa State University Statistical Laboratory and

to the Iowa State University Press for their generous permission

to reprint tables 1, 3, and 4 from their book Statistical Methods,

5th edition. Thanks are also due to Dr. C. I. Bliss of the Connecticut Agricultural Experiment Station, who originally prepared the

material in table 4. I am indebted to Professor Sir Ronald A.

Fisher, F.R.S., Cambridge, and to Dr. Frank Yates, F.R.S., Rothamsted, and to Messrs. Oliver and Boyd Ltd., Edinburgh, for

permission to reprint table 2 from their book Statistical Tables

for Biological, Agricultural, and Medical Research.

FRANK FREESE

Southern Forest Experiment Station
CONTENTS

Page

Basic concepts _ i

Why sample? _ ^

Populations, parameters, and estimates 2

Bias, accuracy, and precision 3

Variables, continuous and discrete _... 5

Distribution functions .-_. 6

Tools of the trade 6

Subscripts, summations, and brackets - 6

Variance 9

Standard errors and confidence limits 10

Expanded variances and standard errors.. 12

Coefficient of variation 13

Covariance 14

Correlation coefficient 15

Independence 16

Variances of products, ratios, and sums 17

Transformations of variables 19

Sampling methods for continuous variables 20

Simple random sampling 20

Stratified random sampling 28

Regression estimators 36

Double sampling 43

Sampling when units are unequal in size (including pps sampling).. 47

Two-stage sampling 50

Two-stage sampling with unequal-sized primaries 57

Systematic sampling 60

Sampling methods for discrete variables 61

Simple random sampling—classification data 61

Cluster sampling for attributes 64

Cluster sampling for attributes—^unequal-sized clusters 67

Sampling of count variables - 68

Some other aspects of sampling 70

Size and shape of sampling units 70

Estimating changes — 73

Design of sample surveys 75
Page

References for additional reading 78

Practice problems in subscript and summation notation 79

Tables —- 82

1. Ten thousand randomly assorted digits 82

2. The distribution of t — 86

3. Confidence intervals for binomial distribution 87

4. Arcsin transformation 89
ELEMENTARY FOREST SAMPLING

This is a statistical cookbook for foresters. It presents some

sampling methods that have been found useful in forestry. No

attempt is made to go into the theory behind these methods. This

has some dangers, but experience has shown that few foresters

will venture into the intricacies of statistical theory until they

are familiar with some of the common sampling designs and

computations.

The aim here is to provide that familiarity. Readers who attain

such familiarity will be able to handle many of the routine sampling problems. They will also find that many problems have been

left unanswered and many ramifications of sampling ignored. It

is hoped that when they reach this stage they will delve into more

comprehensive works on sampling. Several very good ones are

listed on page 78.

BASIC CONCEPTS

Why Sample?

Most human decisions are made with incomplete knowledge. In

daily life, a physician may diagnose disease from a single drop of

blood or a microscopic section of tissue; a housewife judges a

watermelon by its "plug" or by the sound it emits when thumped ;

and amid a bewildering array of choices and claims we select

toothpaste, insurance, vacation spots, mates, and careers with but

a fragment of the total information necessary or desirable for

complete understanding. All of these we do with the ardent hope

that the drop of blood, the melon plug, and the advertising claim

give a reliable picture of the population they represent.

In manufacturing and business, in science, and no less in forestry, partial knowledge is a normal state. The complete census is

rare—^the sample is commonplace. A ranger must advertise timber

sales with estimated volume, estimated grade yield and value, estimated cost, and estimated risk. The nurseryman sows seed whose

germination is estimated from a tiny fraction of the seedlot, and

at harvest he estimates the seedling crop with sample counts in

the nursery beds. Enterprising pulp companies, seeking a source

of raw material in sawmill residue, may estimate the potential

tonnage of chippable material by multiplying reported production

by a set of conversion factors obtained at a few representative

sawmills.

However desirable a complete measurement may seem, there are

several good reasons why sampling is often preferred. In the first

place, complete measurement or enumeration may be impossible.

The nurseryman might b^ somewhat better informed if he knew
2 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

the germinative capacity of all the seed to be sown, but the destructive nature of the germination test precludes testing every

seed. For identical reasons, it is impossible to measure the bending strength of all the timbers to be used in a bridge, the tearing

strength of all the paper to be put into a book, or the grade of

all the boards to be produced in a timber sale. If the tests were

permitted, no seedlings would be produced, no bridges would be

built, no books printed, and no stumpage sold. Clearly where testing is destructive, some sort of sampling is inescapable.

In other instances total measurement or count is not feasible.

Consider the staggering task of testing the quality of all the water

in a reservoir, weighing all the fish in a stream, counting all the

seedlings in a 500-bed nursery, enumerating all the egg masses in

a turpentine beetle infestation, measuring diameter and height of

all the merchantable trees in a 10,000-acre forest. Obviously, the

enormity of the task would demand some sort of sampling

procedure.

It is well known that sampling will frequently provide the essential information at a far lower cost than a complete enumeration.

Less well known is the fact that this information may at times be

more reliable than that obtained by a 100-percent inventory. There

are several reasons why this might be true. With fewer observations to be made and more time available, measurement of the

units in the sample can be and is more likely to be made with greater

care. In addition, a portion of the saving resulting from sampling

could be used to buy better instruments and to employ or train

higher caliber personnel. It is not hard to see that good measurements on 5 percent of the units in a population could provide more

reliable information than sloppy measurements on 100 percent

of the units.

Finally, since sample data can be collected and processed in a

fraction of the time required for a complete inventory, the information obtained may be more timely. Surveying 100 percent of

the lumber market is not going to provide information that is very

useful to a seller if it takes 10 months to complete the job.

Populations, Parameters, and Estimates

The central notion in any sampling problem is the existence of

a population. It is helpful to think of a population as an aggregate

of unit values, where the "unit" is the thing upon which the observation is made, and the "value" is the property observed on that

thing. For example, we may imagine a square 40-acre tract of

timber in which the unit being observed is the individual tree and

the value being observed is tree height. The population is the

aggregate of all heights of trees on the specified forty. The diameters of these same trees would be another population. The cubic

volumes in some particular portion of the stems constitute still

another population.

Alternatively, the units might be defined as the 400 1-chainsquare plots into which the tract could be divided. The cubic

volumes of trees on these plots might form one population. The

board-foot volumes of the same trees would be another populj^-
ELEMENTARY FOREST SAMPLING 3

tion. The number of earthworms in the top 6 inches of soil on

these plots could be still a third population.

Whenever possible, matters will be simplified if the units in

which the population is defined are the same as those to be selected

in the sample. If we wish to estimate the total weight of earthworms in the top 6 inches of soil for some area, it would be best

to think of a population made up of blocks of soil of some specified

dimension with the weight of earthworms in the block being the

unit value. Such units are easily selected for inclusion in the

sample, and projection of sample data to the entire population is

relatively simple. If we think of individual earthworms as the

units, selection of the sample and expansion from the sample to

the population may both be very difficult.

To characterize the population as a whole, we often use certain

constants that are called parameters. The mean value per plot in

a population of quarter-acre plots is a parameter. The proportion

of living seedlings in a pine plantation is a parameter. The total

number of units in the population is a parameter, and so is the

variability among the unit values.

The objective of sample surveys is usually to estimate some

parameter or a function of some parameter or parameters. Often,

but not always, we wish to estimate the population mean or total.

The value of the parameter as estimated from a sample will hereafter be referred to as the sample estimate or simply the estimate.

Bias, Accuracy, and Precision

In seeking an estimate of some population trait, the sampler's

fondest hope is that at a reasonable cost he will obtain an estimate

that is accurate (i.e., close to the true value). Without any help

from sampling theory he knows that if bias rears its insidious

head, accuracy will ñee the scene. And he has a suspicion that

even though bias is eliminated, his sample estimate may still not

be entirely precise. When only a part of the population is measured, some estimates may be high, some low, some fairly close,

and unfortunately, some rather far from the true value.

Though most people have a general notion as to the meaning of

bias, accuracy, and precision, it might be well at this stage to state

the statistical interpretation of these terms.

Bias.—Bias is a systematic distortion. It may be due to some

flaw in measurement, to the method of selecting the sample, or to

the technique of estimating the parameter. If, for example, seedling heights are measured with a ruler from which the first halfinch has been removed, all measurements will be one-half inch too

large and the estimate of mean seedling height will be biased. In

studies involving plant counts, some observers will nearly always

include a plant that is on the plot boundary; others will consistently exclude it. Both routines are sources of measurement bias. In

timber cruising, the volume table selected or the manner in which

it is used may result in bias. A table made up from tall timber will

give biased results when used without adjustment on short-bodied

trees. Similarly, if the cruiser consistently estimates merchantable

height above or below the specifications of the table, volume so
4 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

estimated will be biased. The only practical way to minimize

measurement bias is by continual check of instrumentation, and

meticulous training and care in the use of instruments.

Bias due to method of sampling may arise when certain units

are given a greater or lesser representation in the sample than in

the population. As an elementary example, assume that we are estimating the survival of 10,000 trees planted in 100 rows of 100

trees each. If the sample were selected only from the interior

98 X 98 block of trees in the interest of obtaining a "more representative" picture of survival, bias would occur simply because

the border trees had no opportunity to appear in the sample.

The technique of estimating the parameter after the sample

has been taken is also a possible source of bias. If, for example,

the survival on a planting job is estimated by taking a simple

arithmetic average of the survival estimates from two fields, the

resulting average may be seriously biased if one field is 500 acres

and the other 10 acres in size. A better overall estimate would be

obtained by weighting the estimates for the two fields in proportion to the field sizes. Another example of this type of bias occurs

in the common forestry practice of estimating average diameter

from the diameter of the tree of mean basal area. The latter procedure actually gives the square root of the mean squared diameter, which is not the same as the arithmetic mean diameter unless

all trees are exactly the same size.

Bias is seldom desirable, but it is not a cause for panic. It is

something a sampler may have to live with. Its complete elimination may be costly in dollars, precision, or both. The important

thing is to recognize the possible sources of bias and to weigh the

effects against the cost of reducing or eliminating it. Some of the

procedures discussed in this handbook are known to be slightly

biased. They are used because the bias is often trivial and because

they may be more precise than the unbiased procedures.

Precision and accuracy.—A badly biased estimate may be precise but it can never be accurate. Those who find this hard to

swallow may be thinking of precision as being synonymous with

accuracy- Statisticians being what they are, it will do little good

to point out that several lexicographers seem to think the same

way. Among statisticians accuracy refers to the success of estimating the true value of a quantity; precision refers to the clustering of sample values about their ovm average, which, if biased,

cannot be the true value. Accuracy, or closeness to the true value,

may be absent because of bias, lack of precision, or both.

. ^, ^^rget shooter who puts all of his shots in a quarter-inch

circle m the 10-ring might be considered accurate; his friend who

puts all of his shots in a quarter-inch circle at 12 o'clock in the

b-ring would be considered equally precise but nowhere near as

accurate. An example for foresters might be a series of careful

measurements made of a single tree with a vernier caUper, one

arm of which is not at right angles to the graduated beam.

tJecause the measurements have been carefully made they should

not vary a great deal but should cluster closely about their mean

value : they will be precise. However, as the caliper is not properly
ELEMENTARY FOREST SAMPLING 5

adjusted the measured values will be off the true value (bias) and

the diameter estimate will be inaccurate. If the caliper is properly

adjusted but is used carelessly the measurements may be unbiased

but they will be neither accurate nor precise.

Variables, Continuous and Discrete

Variation is one of the facts of life. It is difficult to say whether

this is good or bad, but we can say that without it there would be

no sampling problems (or statisticians). How to cope with some

of the samp¡ling problems created by natural variation is the

subject of this handbook.

To understand statisticians it is helpful to know their language,

and in this language the term variable plays an active part. A

characteristic that may vary from unit to unit is called a variable.

In a population of trees, tree height is a variable, so are tree diameter, number of cones, cubic volume, and form class. As some trees

may be loblolly pine, some slash pine, and some dawn redwoods,

species is also a variable. Presence or absence of insects, the color

of the foliage, and the fact that the tree is alive or dead are variables also.

A variable that is characterized by being related to some numerical scale of measurement, any interval of which may, if desired, be subdivided into an infinite number of values, is said to

be continuous. Length, height, weight, temperature, and volume

are examples of variables that can usually be labeled continuous.

Qualitative variables and those that are represented by integral

values or ratios of integral values are said to be discrete. Two

forms of discrete data may be recognized : attributes and counts.

In the first of these the individual is classified as having or not

having some attribute ; or, more commonly, a group of individuals

is described by the proportion or percentage having a particular

attribute. Some familiar examples are the proportion of slash pine

seedlings infected by rust, the percentage of stocked milacre quadrats, and the survival percentage of planted seedlings. In the

second form, the individual is described by a count that cannot

be expressed as a proportion. Number of seedlings on a milacre,

number of weevils in a cone, number of sprouts on a stump, and

number of female ñowers on a tree are common examples.

A distinction is made between continuous and discrete variables

because the two types of data may require different statistical

procedures. Most of the sampling methods and computational procedures described in this handbook were developed primarily for

use with continuous variables. The procedures that have been devised for discrete variables are generally more complex. By increasing the number of values that a discrete variable can assume,

however, it is often possible to handle such data by the continuousvariable methods. Thus, germination percentages based on 200 or

more seeds per dish can usually be treated by the same procedures

that would be used for measurement data. The section that begins

on page 61 describes simple random sampling with classification

data and gives some illustrations of how the sampling procedures

for continuous data may be used for classification and count data.
6 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Distribution Functions

A distribution function shows, for a population, the relative frequency with which different values of a variable occur. Knowing

the distribution function, we can say what proportion of the individuals are within certain size limits.

Each population has its own distinct distribution function.

There are, however, certain general types of function that occur

quite frequently. The most common are the normal, binomial, and

Poisson. The bell-shaped normal distribution, familiar to most

foresters, is often encountered in dealing with continuous variables. The binomial is associated with data where a fixed number

of individuals are observed on each unit and the unit is characterized by the number of individuals having some particular attribute. The Poisson distribution may arise where individual units

are characterized by a count having no fixed upper limit, particularly if zero or very low counts tend to predominate.

The form of the distribution function dictates the appropriate

statistical treatment of a set of data. The exact form of the distribution will seldom be known, but some indications may be obtained from the sample data or from a general familiarity with

the population. The methods of dealing with normally distributed

data are simpler than most of the methods that have been developed for other distributions.

Fortunately, it has been shown that, regardless of the distribution which a variable follows, the means of large samples tend to

follow a distribution that approaches the normal and may be

treated by normal distribution methods.

TOOLS OF THE TRADE

Subscripts, Summations, and Braci^ets

In describing the various sampling methods, frequent use will

be made of subscripts, brackets, and summation symbols. Some

beginning samplers will be unhappy about this; others will be

downright mad. The purpose though, is not to impress or confuse

the reader. These devices are, like the more familiar notations

of +» — > and =, merely a concise way of expressing ideas that

would be ponderous if put into conventional language. And like

the common algebraic symbols, using and understanding them is

just a matter of practice.

Subscripts.—The appearance of an Xi, Zjj,, or yumn brings a frown

of annoyance and confusion to the face of many a forester. Yet

interpreting this notation is quite simple. In Xi, the subscript i

means that x can take on different forms or values. Putting in a

particular value of i tells which form or value of x we are concerned with. The i might imply a particular characteristic of an

individual. The term Xi might be the height of the individual, Xo

might be his weight, Xs his age, and so forth. Or the subscript

might imply a particular individual. In this case, Xi could be the

height of the first individual, x^ the height of the second, Xs the
ELEMENTARY FOREST SAMPLING 7

height of the third individual, and so forth. Which meaning is intended will usually be clear from the context.

A variable (say x) will often be identified in more than one

way. Thus, we might want to refer to the age of the second individual or the height of the first individual. This dual classification

is accomplished with two subscripts. In Xij^ the i might identify

the characteristic (for height, i = 1; for weight, i = 2; smà for

age, i = S). The k could be used to designate which individual we

are dealing with. Then, 0^2.7 would tell us that we are dealing with

the weight (i = 2) of the seventh {k = 7) individual. This process can be carried to any length needed. If the individuals in the

above example were from different groups we could use another

subscript (say j) to identify the group. The symbol x^jj^ would

indicate the i^^ characteristic of the k^^ individual of the j'^ group.

Summations.—To indicate that several (say 6) values of a variable (Xi) are to be added together we could write

(^1 -\-X2-\- Xs -}- Xi -\-X5 -\- XQ)

A slightly shorter way of saying the same thing is

(Xi-\-X2-\- ... + Xe)

The three dots (...) indicate that we continue to do the same

thing for all the values from X3 through XQ as we have already

done to Xi and X2.

The same operation can be expressed more compactly by

6

In words this tells us to sum all values of Xi, letting i go from 1

up to 6. The symbol X, which is the Greek letter sigma, indicates

that a summation should be performed. The x tells what is to be

summed and the letter above and below X indicates the limits over

which the subscript i will be allowed to vary.

If all of the values in a series are to be summed, the range of

summation is frequently omitted from the summation sign giving

'^Xi, ^Xi, or sometimes, "^x

«

All of these imply that we would sum all values of Xi.

The same principle extends to variables that are identified by

two or more subscripts. A separate summation sign may be used

for each subscript. Thus, we might have

8 4

i=l j—1

This would tell us to add up all the values of Xij having j from

1 to 4 and i from 1 to 3. Written the long way, this means

(Xi,i + iCi,2 + Xi,3 + ^1.4 + i^2,i + ^2.2

+ i»2,3 + X2,4 + X3,l 4- ^3,2 + %,3 + ^3,4)
8 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

As for a single subscript, when all values in a series are to be

summed, the range of summation may be omitted, and sometimes

a single summation symbol suffices. The above summation might

be symbolized by

S S Xijf S Xij, S S Xiv S i'^íjf or maybe even S x

i 3 i.3

If a numerical value is substituted for one of the letters in the

subscript, the summation is to be performed by letting the letter

subscript vary but holding the other subscript at the specified

value. As an example.

and.

S Xsj = {X3,i -I- ir3,2 + Xs^s + 0:3,4)

3=1

^Xi2 = (aJi.g + 0:2.2 + ^2.2 + ^4.2 + 2^5.2)

Bracketing.—^When other operations are to be performed along

with the addition, some form of bracketing may be used to indicate

the order of operations. For example,

tells us to square each value of Xi and then add up these squared

values. But

(?-)'

tells us to add all the Xi values and then square the sum.

The expression

i 3

says to square each Xij value and then add the squares. But

says that for each value of i we should first add up the Xij over

all values of j. Next, this (s x^j^ jg squared and these squared

sums are added up over all values of i. If the range of / is from

1 to 4 and the range of i is from 1 to 3, then this means
ELEMENTARY FOREST SAMPLING

8/4 \2

+ (i»2,l + fl^.2 + X2,3 + X2,4) 2

+ (i»8.1 + i»3.2 + ^8.3 + 3^3.4) 2

The expression

(2S.«y

would tell us to add up the x^ values over all combinations of i

and / and then square the total. Thus,

(84 \2

ss % ) = (aJi.i + a;i.2 + 0:1,3 + 0:1,4 + 0:2.1 + 0:2.2

»=1 i=l /

+ 0:2,3 + 0:2.4 + 0:3,1 + 0:3,2 + 0:3.3 + 0:3,4)2

Where operations involving two or more different variables are

to be performed, the same principles apply.

s

S XiVt = 0:12/1 + 0:22/2 + 0:32/3

»=1

But,

(S xA (s yij = (xi + X2 + xs) (2/1 + 2/2 + 2/3)

N.B. : It is easily seen but often forgotten that

S Xi^ is not usually equal to (^xA

Similarly,

S Xiyi is not usually equal to (^xÄ ( S 2/í )

Some practice.—If you feel uncomfortable in the presence of

this symbology, try the worked examples on page 79.

Variance

In a stand of trees, the diameters will usually show some variation. Some will be larger than the mean diameter, some smaller,

and some fairly close to the mean. Clearly, it would be informative to know something about this variation. It is not hard to see
10 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

that more observations would be needed to get a good estimate of

the mean diameter in a stand where diameters vary from 2 to 30

inches than where the range is from 10 to 12 inches. The measure

of variation most commonly used by statisticians is the variance.

The variance of individuals in a population is a measure of the

dispersion of individual unit values about their mean. A large

variance indicates wide dispersion, a small variance indicates little

dispersion. The variance of individuals is a population characteristic (a parameter). Very rarely will we know the population

variance. Usually it must be estimated from the sample data.

For most types of forest measurement data, the estimate of the

variance from a simple random sample is given by

Q2 — »Hi

' ~ {n-D

Where : s^ = Sample estimate of the population variance.

yi — The value of the i^^ unit in the sample.

y = The arithmetic mean of the sample, i.e.,

n

S yi

mi, ^ "r ^^® number of units observed in the sample.

Ihough It may not appear so, computation of the sample variance IS simplified by rewriting the above equation as

52 = i^l-

(â ^'T

(M-1)

8 Ä°f^t^lt^® observations on three units with the values 7,

», and 12. For this sample our estimate of the variance is

(72 + 82 + 122) — !^Ú1

«2 = 3 257-243 „

2 - 2 = ^

forest m^^rÄ^**"'*' ^ i^ *^'»"i*'^ to the survivors of most

anee K if«Ar Tí.^^^' '' T^^^l^ ^^^ «1«a»-e root of the variance. It is symbolized by s, and in the above example would be

estimated as s = y? = 2.6458.

Standard Errors and Confidence Limits

subfe^cVto^ S"^^nil""^ '" a population, sample estimates are subject to variation. The mean diameter of a stand as estimated
ELEMENTARY FOREST SAMPLING 11

from a sample of 3 trees will seldom be the same as the estimate

that would have been obtained from other samples of 3 trees. One

estimate might be close to the mean but a little high. Another

might be quite a bit high, and the next might be below the mean.

The estimates vary because different individual units are observed

in the different samples.

Obviously, it would be desirable to have some indication of how

much variation might be expected among sample estimates. An

estimate of mean tree diameter that would ordinarily vary between 11 and 12 inches would inspire more confidence than one

that might range from 6 to 18 inches.

The previous section discussed the variance and the standard

deviation (standard deviation = Vvariance) as measures of the

variation among individuals in a population. Measures of the

same form are used to indicate how a series of estimates might

vary. They are called the variance of the estimate and the

standard error of estimate (standard error of estimate =

Vvariance of estimate). The term, standard error of estimate, is

usually shortened to standard error when the estimate referred

to is obvious.

The standard error is merely a standard deviation, but among

estimates rather than among individual units. In fact, if several

estimates were obtained by repeated sampling of a population, the

variance and standard error of these estimates could be computed

from the equations given in the previous section for the variance

and standard deviation of individuals. But repeated sampling is

unnecessary; the variance and standard error can be obtained

from a single set of sample units. Variability of an estimate depends on the sampling method, the sample size, and the variability

among the individual units in the population, and these are the

pieces of information needed to compute the variance and standard error. For each of the sampling methods described in this

handbook, the procedure for computing the standard error of

estimate will be given.

Computation of a standard error is often regarded as an unnecessary frill by some self-styled practical foresters. The fact is,

however, that a sample estimate is almost worthless without some

indication of its reliability.

Given the standard error, it is possible to establish limits that

suggest how close we might be to the parameter being estimated.

These are called confidence limits. For large samples we can take

as a rough guide that, unless a l-in-3 chance has occurred m

sampling, the parameter will be within one standard error of the

estimated value. Thus, for a sample mean tree diameter of 16

inches with a standard error of 1.5 inches, we can say that the

true mean is somewhere within the limits 14.5 to 17.5 inches. In

making such statements we will, over the long run, be right an

average of two times out of three. One time out of three we will,

because of natural sampling variation, be wrong. The values

given by the sample estimate plus or minus one standard error

are called the 67-percent confidence limits. By spreading the limits

we can be more confident that they will include the parameter.
12 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Thus, the estimate plus or minus two standard errors will give

limits that will include the parameter unless a l-in-20 chance has

occurred. These are called the 95-percent confidence limits. The

99-percent confidence limits are defined by the mean plus or minus

2.6 standard errors. The 99-percent confidence limits will include

the parameter unless a 1-in-lOO chance has occurred.

It must be emphasized that this method of computing confidence

limits will give valid approximations only for large samples. The

definition of a large sample depends on the population itsetf, but

in general any sample of less than 30 observations would not

qualify. Some techniques of computing confidence limits for small

samples will be discussed for a few of the sampling methods.

Expanded Variances and Standard Errors

Very often an estimate will be multiplied by a constant to put

it in a more meaningful form. For example, if a survey has been

made using one-fifth acre plots and the mean volume per plot

computed, this estimate would be multiplied by 5 in order to put

the estimated mean on a per acre basis. Or, for a tract of 800

acres the mean volume per fifth-acre plot would be multiplied by

4,000 (the number of one-fifth acres in the tract) in order to

estimate the total volume.

Since expanding a variable in this way must also expand its

variability, it will be necessary to compute a variance and standard error for these expanded values. This is easily done. If the

variable x has variance s^ and this variable is multiplied by a constant (say k), the product {kx) will have a variance of k^s"^.

Suppose the estimated mean volume per one-fifth acre plot is

1,400 board feet with a variance of 2,500 board feet (giving a

standard error of V2,500 = 50 board feet). The mean volume

per acre is

Mean volume per acre = 5 (1,400) = 7,000 board feet

and the variance of this estimate is

Variance of mean volume per acre = (5^) (2,500) = 62,500

The standard error of the mean volume per acre would be

V Variance of mean volume per acre = 250 board feet

Note that if the standard deviation (or standard error) of x is

s, then the standard deviation (or standard error) of kx is merely

Ks, So, m the above case, since the standard error of the estimated

mean volume per fifth-acre plot is 50, the standard error of the

mean volume per acre is (5) (50) m 250.

This is a simple but very important rule and anyone who will

be dealing with sample estimates should master it.

Variables may also be expanded by the addition of a constant.

Hixpansion of this type does not affect variability and requires no

adjustment of the variance or standard errors. Thus if

zr=zx \-k
ELEMENTARY FOREST ÖAMPLING IS

where a; is a variable and k a constant, then

o 2 — <j 2

This situation arises where for computational purposes the data

have been coded by the subtraction of a constant. The variance

and standard error of the coded values are the same as for the

uncoded values. Given the three observations 127, 104, and 114

we could, for ease of computation, code these values by subtracting 100 from each, to make 27, 4, and 14. The variance of the

coded values is

(272 + 42 + 142) _ iM!

s^ = 2 = 1^^

which is the same as the variance of the original values

(1272 +1042 + 1142) -i?^

:133

Coefficient of Variation

The coefficient of variation (C) is the ratio of the standard deviation to the mean. For a sample with a mean^ of ^ = 10 and a

standard deviation of s = 4 we would estimate the coefficient of

variation as

s 4

C = - = jjT = OA or 40 percent

X J.U

Variance, our measure of variability among units, is often related to the mean size of the units; large items tend to have a

larger variance than small items. For example, the variance in a

population of tree heights would be larger than the variance of

the heights of a population of foresters. The coefficient of variation puts the expression of variability on a relative basis. The

population of tree heights might have a standard deviation of 4.4

feet while the population of foresters might have a standard deviation of 0.649 foot. In absolute units, the trees are more variable

than the foresters. But, if the mean tree height is 40 feet and the

mean height of the foresters is 5.9 feet, the two populations would

have the same relative variability. They would both have a coefficient of variation oí C = 0.11.

Variance also depends on the measurement units used. The

standard deviation of foresters' heights was 0.649 foot. Had the

heights been measured in inches, the standard deviation would

have been 12 times as large (If z = 12ir s^ = 12sJ or 7.788

inches. But the coefficient of variation would be the same regardless of the unit of measure. In either case, we would have

^ s 0.649 foot 7.788 inches ^ ^ ^ ^^ .. ^o^n^r,+ C = — = —^r-;r-^—r = i^Tm—:—Ï— = 0.11 or 11 perccnt X 5.9 feet 70.8 mches

1 The sample mean of a variable x is frequently symbolized by x.
14 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

In addition to putting variabilities on a comparable basis, the

coefficient of variation simplifies the job of estimating and remembering the degree of variability of different populations. In

many of the populations with which foresters deal, the coefficient

of variation is approximately 100 percent. Because it is often

possible to guess at the size of the population mean, we can readily

estimate the standard deviation. Such information is useful in

planning a sample survey.

Covariance

In some sampling methods measurements are made on two or

more characteristics for each sample unit. In measuring forage

production, for example, we might get the green weight of the

grass clipped to a height of 1 inch from a circular plot 1 foot in

diameter. Later we might get the ovendry weight of the same

sample.

Covariance is a measure of how two variables vary in relationship to each other (covariability). Suppose the two variables are

labeled y and x. If the larger values of y tend to be associated

with the larger values of x, the covariance will be positive. If the

larger values of y are associated with the smaller values of x, the

covariance will be negative. When there is no particular association of y and x values, the covariance approaches zero. Like

the variance, the covariance is a population characteristic—a

parameter.

For simple random samples, the formula for the estimated covariance {Sa,y) of X and y is

S {Xi — x){yi — \

O^ZTi)

Computation of the sample covariance is simplified by rewriting

the formula

S Xiyi- (É '') (É ')

n — 1

Suppose that a sample of n = 6 units has produced the foUowmg X and y values :

i 1

Vi 2

Xi 12

2 3 4 5 6 Totals

12 7 14 11 8 54

4 10 3 6 7 42
ELEMENTARY POEEST SAMPLING 15

Then,

^ (2) (12) + (12) (4) + . .. + (8) (7) -(ÍMH42)\

The negative value indicates that the larger values of y tend

to be associated with the smaller values of x.

Correlation Coefficient

The magnitude of the covariance, like that of the variance, is

often related to the size of the unit values. Units with large values

of X and y tend to have larger covariance values than units with

smaller x and y values. A measure of the degree of linear association between two variables that is unaffected by the size of the

unit values is the simple correlation coefficient. A sample-based

estimate (r) of the correlation coefficient is

Covariance of x and y Sa,v

V (Variance of a;) (Variance ofy) VT^^TTVT

The correlation coefficient can vary between —1 and -j-l. As in

covariance, a positive value indicates that the larger values of y

tend to be associated with the larger values of x, A negative value

indicates an association of the larger values of y with the smaller

values of x. A value close to +1 or —1 indicates a strong linear

association between the two variables. Correlations close to zero

suggest that there is little or no linear association.

For the data given in the discussion of covariance we found

Saiy = —14.4. For the same data, the sample variance of x is

s/ = 12.0, and the sample variance of y is s/ _ ig,4. Then the

estimate of the correlation between y and x is

—14 4 —14 4

^^'^ ^^•^. = -0.969

" V (12.0) (18.4) 14.86

The negative value indicates that as x increases y decreases, while

the nearness of r to —1 indicates that the linear association is very

close.

An important thing to remember about the correlation coefficient is that it is a measure of the linear association between two

variables. A value of r close to zero does not necessarily mean

that there is no relationship between the two variables. It merely

means that there is not a good linear (straight-line) relationship.

There might actually be a strong nonlinear relationship.

It must also be remembered that the correlation coefficient computed from a set of sample data is an estimate, just as the sample

mean is an estimate. Like the sample, the reliability of a córrela-
16 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

tion coefficient increases with the sample size. Most statistics

books have tables that help in judging the reliability of a sample

correlation coefficient.

Independence

When no relationship exists between two variables they are said

to be independent; the value of one variable tells us absolutely

nothing about the value of the other. The common measures of

independence (or lack of it) are the covariance and the correlation coefficient. As previously noted, when there is little or no

association between the values of two variables, their covariance

and correlation approach zero (but keep in mind that the converse

is not necessarily true; a zero correlation does not prove that

there is no association but only indicates that there is no strong

linear relationship).

Completely independent variables are rare in biological populations, but many variables are very weakly related and may be regarded as independent. As an example, the annual height growth

of pole-sized loblolly pine dominants is relatively independent of

the stand basal area within fairly broad limits (say 50 to 120

square feet per acre). There is also considerable evidence that

periodic cubic volume growth of loblolly pine is poorly associated

with (i.e., almost independent of) stand basal area over a fairly

wide range.

The concept of independence is also applied to sample estimates.

In this case, however, the independence (or lack of it) may be

«^e to the sampling method as well as to the relationship between

the basic variables. For discussion purposes, two situations may

be recognized:

Two estimates have been made of the same parameter.

Estimates have been made of two different parameters.

In the first situation, the degree of independence depends entirely on the method of sampling. Suppose that two completely

separate surveys have been made to estimate the mean volume per

^^î"^ A\.?: ^^^^^^ stand. Because different sample plots are involved, the estimates of mean volume obtained from these surveys

would be regarded as statistically independent. But suppose an

estimate has been made from one survey and then additional

sample plots are selected and a second estimate is made using the

plot data from both the first and second surveys. Since some of

the same observations enter both estimates, the estimates would

not be independent. In general, two estimates of a single param-

•l ^^ ml* Independent if some of the same observations are used

m Doth. Ihe degree of association will depend on the proportions

of observations common to the two estimates.

In the second situation (estimates of two different parameters)

wAtf^îC independence may depend on both the sampling

T? ZLr^T • *if ^^¥^^ ""^ association between the basic variables,

iofî^of^i^u ^^* ^i^ ^^^'^ diameter of a population of trees were

^l^^^^yi \TÎÎ'^^? selecting a number of individual trees and

^fSio^ both the height and diameter of each tree, the two

estimates would not be independent. The relationship between
ELEMENTARY FOREST SAMPLING 17

the two estimates (usually measured by their covariance or correlation) would, in this case, depend on the degree of association

between the height and diameter of individual trees. On the other

hand, if one set of trees were used to estimate mean height and

another set were selected for estimating mean diameter, the two

estimates would be statistically independent even though height

and diameter are not independent when measured on the same tree.

A measure of the degree of association (covariance) between

two sample estimates is essential in the evaluation of the sampling

error for several types of surveys. For the sampling methods described in this handbook, the procedure for computing the covariance of two estimates will be given when needed.

Variances of Products, Ratios, and Sums

In a previous section, we learned that if a quantity is estimated

as the product of a constant and a variable (say Q = kz, where

fc is a constant and z is a variable) the variance of Q will be

SQ2 _ Jc^sJ^, Thus, if we wish to estimate the total volume of a

stand, we would multiply the estimated mean per unit (y, a variable) by the total number of units (N, a constant) in the population. The variance of the estimated total will be N%^. Its standard deviation (or standard error) would be the square root of its

variance or Nsy.

The variance of a product.—^In some situations the quantity in

which we are interested will be estimated as the product of two

variables and a constant. Thus,

Qi = kzw

where : fc = a constant and

z and w = variables having variances s/ and sj^ and

covariance Sgy,

For large samples, the variance of Qi is estimated by

V = Ox^($ + ^+-

As an example of such estimates, consider a large forest survey

project which uses a dot count on aerial photographs to estimate

the proportion of an area that is in forest (p), and a ground

cruise to estimate the mean volume per acre (v) oí forested land.

To estimate the forested acreage, the total acreage (N) in the

area is multiplied by the estimated proportion forested. This in

turn is multiplied by the mean volume per forested acre to give

the total volume. In formula form

Total volume = N (p) (v)

Where : N = The total acreage of the area (a known constant).

p = The estimated proportion of the area that is forested.

V = The estimated mean volume per forested acre.
18 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

The variance of the estimated total volume would be

If the two estimates are made from separate surveys, they are

assumed to be independent and the covariance set equal to zero.

This would be the situation here where p is estimated from a

photo dot count and v from an independently selected set of

ground locations. With the covariance set equal to zero, the variance formula would be

'=(^^(^)w)X¥+¥)

Variance of a ratio.—^In other situations, the quantity we are

interested in will be estimated as the ratio of two estimates multiplied by a constant. Thus, we might have

w

For large samples, the variance of Q2 can be approximated by

^«2 -Q2 \_-^-^^--^\

This formula comes into use with the ratio-of-means estimator

described m the section on regression estimators.

Variance of a sí¿m.—Sometimes we might wish to use the sum

of two or more variables as an estimate of some quantity. With

two variables we might have

Q3 = kiXx 4- k2X2

where: ki and k2 = constants

Xi and X2 =z variables having variance Si^ and Sz^

and covariance S12

The variance of this estimate is

SQS^ = kiW -f k2^S2^ + 2kik2Si2

If we measure the volume of sawtimber (x) and the volume of

poletimber (^) on the same plots (and in the same units of measure) and find the mean volumes to be x and ^, with variances

s^ and 5^2 and covariance s^, then the mean total volume in polesize and larger trees would be

Mean total volume = :g -j- ^

The variance of this estimate is

s2 = s/^s^2^2s¿ij
ELEMENTARY FOREST SAMPLING 19

The same result would, of course, be obtained by totaling the x

and y values for each plot and then computing the variance of

the totals.

This formula is also of use where a weighted mean is to be computed. For example, we might have made sample surveys of two

tracts of timber.

Tract 1

Size = 3,200 acres

Estimated mean volume per acre = 4,800 board feet

Variance of the mean = 112,500 board feet

Tract 2

Size = 1,200 acres

Estimated mean volume per acre = 7,400 board feet

Variance of the mean = 124,000 board feet

In combining these two means to estimate the overall mean volume

per acre we might want to weight each mean by the tract size

before adding and then divide the sum of the weighted means by

the sum of the weights (this is the same as estimating the total

volume on both tracts and dividing by the total acreage to get the

mean volume per acre). Thus,

_ 3200(4800) + 1200(7400)

^ ~ (3200 4- 1200)

=(S)'«««) + (S) ""<»=»»

Because the two tract means were obtained from independent

samples, the covariance between the two estimates is zero, and the

variance of the combined estimate would be

z= (3200)^112,500) + (1200)M124,000)

(4400)2

= 68,727.

The general rule for the variance of a sum is if

Q = kyXi -\- kzXz + . . . + KXn

where : kt = constants

Xi = variables with variances s/ and covariance Si^,

then

Sg2 _ ^^25^2 ^ ^^W + ... + KV + 2klk2Sl2

+ 2kik3Si3 + . . . + 2kn.l knS^n.l)n

Transformation of Variables

Many of the procedures described in this handbook imply certain assumptions about the nature of the variable being studied.
20 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

When a variable does not fit the assumptions for a particular procedure some other method must be used or the variable must be

changed (transformed).

One of the common assumptions is that variability is independent of the mean. Some variables (e.g., those that follow a

binomial or Poisson distribution) tend to have a variance that is

in some way related to the mean—populations with large means

often having large variance. In order to use procedures that assume that there is no relationship between the variance and the

mean, these variables are frequently transformed. The transformation, if properly selected, puts the original data on a scale in

which its variability is independent of the mean. Some common

transformations are the square root, arcsin, and logarithm. The

arcsin transformation is illustrated on page 66.

If a method assumes that there is a linear relationship between

two variables, it is often necessary to transform one or both of

the variables so that they satisfy this assumption. A variable may

also be transformed to convert its distribution to the normal on

which many of the simpler statistical methods are based.

The amateur sampler will do well to seek expert advice when

transformations are being considered.

Finally, it should be noted that transformation is not synonymous with coding, which is done to simplify computation. Nor is

it a form of mathematical hocus-pocus aimed at obtaining answers

that are in agreement with preconceived notions.

SAMPLING METHODS FOR CONTINUOUS VARIABLES

Simple Random Sampling

All of the sampling methods to be described in this handbook

have their roots in simple random sampling. Because it is basic,

the method will be discussed in greater detail than any of the

other procedures.

The fundamental idea in simple random sampling is that, in

^ÎÎ^^fi^Ç ^ sample of n units, every possible combination of n units

should have an equal chance of being selected. This is not the

same as requirmg that every unit in the population have an equal

chance of being selected. The latter requirement is met by many

iorms of restricted randomization and even by some systematic

selection methods.

Giving every possible combination of n units an equal chance

pi appearing m a sample of size n may be difficult to visualize but

IS easily accomplished. It is only necessary to be sure that at any

stage oí the sampling the selection of a particular unit is in no

way influenced by the other units that have been selected. To state

It m another way, the selection of any given unit should be completely independent of the selection of all other units. One way

to do this IS to assign every unit in the population a number and

^^ Q^x^^A^ Vi numbers from a table of random digits (table 1,

p. »Z). Or, the numbers can be written on some equal-sized disks

or slips of paper which are placed in a bowl, thoroughly mixed.
ELEMENTARY FOREST SAMPLING 21

and then drawn one at a time. For units such as individual tree

seeds, the units themselves may be drawn at random.

The units may be selected with or without replacement. If selection is with replacement, each unit is allowed to appear in the

sample as often as it is selected. In sampling without replacement,

a particular unit is allowed to appear in the sample only once.

Most forest sampling is without replacement. As will be shown

later, the procedure for computing standard errors depends on

whether sampling was with or without replacement.

Sample selection.—The selection method and computations may

be illustrated by the sampling of a 250-acre plantation. The objective of the survey was to estimate the mean cordwood volume per

acre in trees more than 5 inches d.b.h. outside bark. The population and sample units were defined to be square quarter-acre plots

with the unit value being the plot volume. The sample was to consist of 25 units selected at random and without replacement.

The quarter-acre units were plotted on a map of the plantation

and assigned numbers from 1 to 1,000. From a table of random

digits, 25 three-digit numbers were selected to identify the units

to be included in the sample (the number 000 was associated with

the plot numbered 1,000). No unit was counted in the sample more

than once. Units drawn a second time were rejected and an alternative unit was randomly selected.

The cordwood volumes measured on the 25 units were as

follows :

7 10 7 4 7

8 8 8 7 5

2 6 9 7 8

6 7 11 8 8

7 3 8 7 7

Total =:175

Estimates,—^If the cordwood volume on the i^^ sampling unit is

designated Vt, the estimated mean volume (y) per sampling unit is

n

.-. _ il^' _ 7 + 8-f-2 + ...-f7 _ 175

^ - ~~ir - 25 ~ 25

= 7 cords per quarter-acre plot.

The mean volume per acre would, of course, be 4 times the mean

volume per quarter-acre plot, or 28 cords.

As there is a total of N = 1,000 quarter-acre units in the 250-

acre plantation, the estimated total volume (7) in the plantation would be

Y=zNy= (1,000) (7) = 7,000 cords.

Alternatively,

f z=z (28 cords per acre) (250 acres) = 7,000 cords.
22 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

Standard errors,—A first step in computing the standard error

of estimate is to make an estimate (Sy^) of the variance of individual values of y.

(È'^ï

., ,?. ^'^- «'

"' - (« -1)

In this example,

sJ

(72+ 82+ ...+72) -

(175)2

25

(25 - 1)

^ 1.317 - 1,225 ^ 3 3333 ^^^^^

24

When sampling is without replacement the standard error of

the mean (s^) for a simple random sample is

•.=V¥('-*)

where : N = total number of sample units in the entire population,

n = number of units in the sample.

For the plantation survey,

** = V^l-ïSô) = V (.1533) (.975)

= 0.387 cord

This is the standard error for the mean per quarter-acre plot. By

the rules for the expansion of variances and standard errors, the

standard error for the mean volume per acre will be (4) (0.387)

= 1.548 cords.

Similarly, the standard error for the estimated total volume

(sf) will be

Sf = NSi = (1,000) (.387) = 387 cords.

Sampling with replacement.—In the formula for the standard

error of the mean, the term (1 —jr) is known as the finite population correction or fpc. It is used when units are selected without replacement. If units are selected with replacement, the fpc
ELEMENTARY FOREST SAMPLING 23

is omitted and the formula for the standard error of the mean

becomes

8, = ^.

Even when sampling is without replacement the sampling fraction (n/N) may be extremely small, making the fpc very close to

unity. If n/N is less than 0.05, the fpc is commonly ignored and

the standard error computed from the shortened formula.

Confidence limits for large samples.—By itself, the estimated

mean of 28 cords per acre does not tell us very much. Had the

sample consisted of only 2 observations we might conceivably have

drawn the quarter-acre plots having only 2 and 3 cords, and the

estimated mean would be 10 cords per acre. Or if we had selected

the plots with 10 and 11 cords, the mean would be 42 cords per

acre.

To make an estimate meaningful it is necessary to compute

confidence limits that indicate the range within which we might

expect (with some specified degree of confidence) to find the

parameter. As was discussed in the chapter on standard errors,

the 95-percent confidence limits for large samples are given by

Estimate zt 2 (Standard Error of Estimate)

Thus the mean volume per acre (28 cords) that had a standard

error of 1.548 cords would have confidence limits of

28 ± 2 (1.548) = 24.90 to 31.10 cords per acre.

And the total volume of 7,000 cords that had a standard error of

387 cords would have 95-percent confidence limits of

7,000 zt 2 (387) = 6,226 to 7,774 cords.

Unless a l-in-20 chance has occurred in sampling, the population mean volume per acre is somewhere between 24.9 and 31.1

cords, and the true total volume is between 6,226 and 7,774 cords.

Because of sampling variation, the 95-percent confidence limits

will, on the average, fail to include the parameter in 1 case out

of 20. It must be emphasized, however, that these limits and the

confidence statement take account of sampling variation only.

They assume that the plot values are without measurement error

and that the sampling and estimating procedures are unbiased

and free of computational mistakes. If these basic assumptions are

not valid, the estimates and confidence statements may be nothing

more than a statistical hoax.

Confidence limits for small samples.—Ordinarily, large-sample

confidence limits are not appropriate for samples of less than 30

observations. For smaller samples the proper procedure depends

on the distribution of the unit values in the parent population, a

subject that is beyond the scope of this handbook. Fortunately,

many forest measurements follow the bell-shaped normal distribution, or a distribution that can be made nearly normal by transformation of the variable.
24 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

For samples of any size from normally distributed populations,

Student's t value can be used to compute confidence limits. The

general formula is

Estimate ± (t) (Standard Error of Estimate).

The values of t have been tabulated (table 2, page 86). The particular value of t to be used depends on the degree of confidence

desired and on the size of the sample. For 95-percent confidence

limits, the t values are taken from the column for a probability

of .05. For 99-percent confidence limits, the t value would come

from the .01 probability column. Within the specified colunms, the

appropriate t for a simple random sample of n observations is

found in the row for (n — 1) df's (degrees of freedom^). For a

simple random sample of 25 observations the t value for computing the 95-percent confidence limits will be found in the .05 column

and the 24 df row. This value is 2.064. Thus, for the plantation

survey that showed a mean per-acre volume of 28 cords and a

standard error of the mean of 1.548 cords, the small-sample 95-

percent confidence limits would be

28 ± (2.064) (1.548) = 24.80 to 31.20 cords

The same t value is used for computing the 95-percent confidence

limits on the total volume. As the estimated total was 7,000 cords

with a standard error of 387 cords, the 95-percent confidence

limits are

7,000 ± (2.064) (387) = 6,201 to 7,799 cords.

Size of sample.—In the example illustrating simple random

sampling, 25 units were selected. But why 25? Why not 100? Or

10? All too often the number depends on the sampler's view of

what looks about right. But there is a somewhat more objective

solution. That is to take only the number of observations needed

to give the desired precision.

In planning the plantation survey, we could have stated that

unless a l-in-20 chance occurs we would like our sample estimate

of the mean to be within ±: E cords of the population mean. As

the small-sample confidence limits are computed as y it t (Sy), this

is equivalent to saying that we want

tisç) = E

For a simple random sample

«-V¥(^-F)

2 In this handbook the expression "degrees of freedom" refers to a parameter

in the distribution of Student's t. When a tabular value of t is required, the

number of degrees of freedom (df's) must be specified. The expression is not

easily explained in nonstatistical language. One definition is that the df*s are

equal to the number of observations in a sample minus the number of independently estimated parameters used in calculating the sample variance. Thus,

in a simple random sample of n observations the only estimated parameter

needed in calculating the sample variance is the mean (a?), and so the df's

would be (n — 1).
ELEMENTARY FOREST SAMPLING 25

Substituting for Sy in the first equation we get

Rewritten in terms of the sample size (n) this becomes

To solve this relationship for n, we must have some estimate

(s/) of the population variance. Sometimes the information is

available from previous surveys. In the illustration, we found

Sy^ = 3.83, a value which might be taken as representative of the

variation among quarter-acre plots in this or similar populations.

In the absence of this information, a small preliminary survey

might be made in order to obtain an estimate of the variance.

When, as often happens, neither of these solutions is feasible, a

very crude estimate can be made from the relationship

-(?)

where : R = estimated range from the smallest to the largest unit

value likely to be encountered in sampling.

For the plantation survey we might estimate the smallest y-value

on quarter-acre plots to be 1 cord and the largest to be 10 cords.

As the range is 9, the estimated variance would be

■■ = {iy =5.06

This approximation procedure should be used only when no other

estimate of the variance is available.

Having specified a value of E and obtained an estimate of the

variance, the last piece of information we need is the value of t.

Here we hit somewhat of a snag. To use t we must know the

number of degrees of freedom. But, the number of df's must be

(n ~ 1) and n is not known and cannot be determined without

knowing t

An iterative solution will give us what we need, and it is not as

difficult as it sounds. The procedure is to guess at a value of n,

use the guessed value to get the degrees of freedom for t, and

then substitute the appropriate t value in the sample-size formula

to solve for a first approximation of n. Selecting a new n somewhere between the guessed value and the first approximation, but

closer to the latter, we compute a second approximation. The process is repeated until successive values of n are the same or only

slightly different. Three trials usually suffice.

To illustrate the process, suppose that in planning the plantation survey we had specified that, barring a 1-in-lOO chance, we

would like the estimate to be vdthin 3.0 cords of the true mean
26 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

volume per acre. This is equivalent to E = 0.75 cord per quarteracre. From previous experience, we estimate the population variance among quarter-acre plots to be s/ = 4, and we know that

there is a total of N = 1,000 units in the population. To solve for

n, this information is substituted in the sample-size formula given

on page 25.

(0.75)2 ^

(Í2) (4) "^ 1,000

We will have to use the t value for the .01 probability level, but

we do not know how many degrees of freedom t will have without

knowing n. As a first guess, we can try n = 61; then the value

of t with 60 degrees of freedom at the .01 probability level is

t = 2.66. Thus, the first approximation will be

^'- (0.752) 1 - .5625

(2.662) (4) T 1^000 (7.0756) (4) ^ 1.000

= 47.9

A second guessed value for n would be somewhere between 61 and

48, but closer to the computed value. We might test n = 51, for

which the value of í (50 df*s) at the .01 level is about 2.68, whence

712 =

.5625 1

(7.1824) (4) "^ 1,000

= 48.6

The desired value is somewhere between 51 and 48.6 but much

closer to the latter. Because the estimated sample size is, at best,

only a good approximation, it is rather futile to strain on the computation of n. In this case we would probably settle on n = 50,

a value that could have been easily guessed after the first approximation was computed.

If the sampling fraction-^ is likely to be small (say, less than

0.05), the finite-population correction il — ^ ) may be ignored

in the estimation of sample size and the formula simplified to

This formula is also appropriate in sampling with replacement.

In the previous example the simplified formula gives an estimated

sample size oí n = 51.

The short formula is frequently used to get a first approximation of n. Then, if the sample size indicated by the short formula
ELEMENTARY FOREST SAMPLING 27

is a considerable proportion (say over 10 percent) of the number

of units in the population and sampling will be without replacement, the estimated sample size is recomputed with the long

formula.

Ejfect of plot size on variance.—^In estimating sample size, the

effect of plot size and the scale of the unit values on variance must

be kept in mind. In the plantation survey a plot size of one-quarter

acre was selected and the variance among plot volumes was estimated to be s^ = 4. This is the variance among volumes per

quarter-acre. Because the desired precision was expressed on a

per-acre basis it was necessary to modify either the precision

specification or s^ to get them on the same scale. In the example,

s^ was used without change and the desired precision was divided

by 4 to put it on a quarter-acre basis. The same result could have

been obtained by leaving the specified precision unchanged and

putting the variance on a per-acre basis. Since the quarter-acre

volumes would be multiplied by 4 to put them on a per-acre basis,

the variance of quarter-acre volumes should be multiplied by 16.

(Remember : If a; is a variable with variance s^, then the variance

of a variable z = kx is k^s^).

Plot size has an additional effect on variance. At the same scale

of measurement, small plots will almost always be more variable

than large ones. The variance in volume per acre on quarter-acre

plots would be somewhat larger than the variance in volume per

acre on half-acre plots, but slightly smaller than the variance in

volume per acre of fifth-acre plots. Unfortunately, the relation of

plot size to variance changes from one population to another.

Large plots tend to have a smaller variance because they average

out the effect of clumping and holes. In very uniform populations,

changes in plot size have little effect on variance. In nonuniform

populations the relationship of plot size to variance will depend

on how the sizes of clumps and holes compare to the plot sizes.

Experience is the best guide as to the effect of changing plot size

on variance. Where neither experience nor advice is available, a

very rough approximation can be obtained by the rule :

If plots of size Pi have a variance Si^ then, on the same scale

of measurement, plots of size Pa will have a variance roughly

equal to

S2^ = Si2 ^/PJP¡

Thus, if the variance in cordwood volume per acre on quarter-acre

plots is Si^ = 61, the variance in cordwood volume per acre on

tenth-acre plots will be roughly

61 VO.25/0.10 = 96

The same results will be obtained without worry about the scale

of measurement if the squared coefficients of variation (C^) are

used in place of the variances. The formula would then be
28 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Practice problem.—^A survey is to be made to estimate the mean

board-foot volume per acre in a 200-acre tract. Barring a l-in-20

chance, we would like the estimate to be within 500 board feet of

the population mean. Sample plots will be one-fifth acre. A survey

in a similar tract showed the standard deviation among quarteracre plot volumes to be 520 board feet. What size sample will be

needed?

Problem Solution:

The variance among quarter-acre plot volumes is 520^ = 270,-

400. For quarter-acre volumes expressed on a per-acre basis the

variance would be

Si2 = (42) (270,400) = 4,326,400

The estimated variance among fifth-acre plot volumes expressed

on a per-acre basis would then be

''=''^¥¡=''^^''^''^0.20 Vo.2

= (4,326,400) (1.118)

= 4,836,915

The population size is iV = 1,000 fifth-acre plots.

If as a first guess n = 61, the t value at the .05 level with 60

degrees of freedom is 2.00. The first computed approximation of

n is

^-= (500)^' , —-^^-^

(4) (4,836,915) "^ 1,000

The correct solution is between 61 and 71.8 but much closer to the

computed value. Repeated trials will give values between 71.0 and

71.8. The sample size (n) must be an integral value and, because

71 is too small, a sample of n = 72 observations would be required for the desired precision.

Stratified Random Sampling

Often we have knowledge of a population which can be used to

increase the precision or usefulness of our sample. Stratified random sampling is a method that takes advantage of certain types

of mformation about the population.

In stratified random sampling, the units of the population are

grouped together on the basis of similarity of some characteristic.

Each group or stratum is then sampled and the group estimates

are combined to give a population estimate.

In sampling a forest, we might set up strata corresponding to

the major timber types, make separate sample estimates for each

type, and then combine the type data to give an estimate for the

entire population. If the variation among units within tsrpes is
ELEMENTARY FOREST SAMPLING 29

less than the variation among units that are not in the same type,

the population estimate will be more precise than if sampling had

been at random over the entire population.

The sampling and computational procedures can be illustrated

with data from a cruise made to estimate the mean cubic-foot

volume per acre on an 800-acre forest. On aerial photographs the

tract was divided into three strata corresponding to three major

forest types ; pine, bottom-land hardwoods, and upland hardwoods.

The boundaries and total acreage of each type were known. Ten

one-acre plots were selected at random and without replacement

in each stratum.

Stratum

I. Pine

II. Bottom-land hardwoods

Observations

570 510 600

640 590 780 Total == 6,100

480 670 700

560

520 630 810

710 760 580 Total == 7,370

770 890 860

840

420 540 320

210 180 270 Total == 3,040

290 260 200

350

III. Upland hardwoods

Estimates.—^The first step in estimating the population mean

per unit is to compute the sample mean (yn) for each stratum.

The procedure is the same as for the mean of a simple random

sample.

yI =z 6,100/10 = 610 cubic feet per acre for the pine type

yii = 7,370/10 = 737 cubic feet per acre for bottom-land hardwoods

yiii = 3,040/10 = 304 cubic feet per acre for upland hardwoods

The mean of a stratified sample (fst) is then computed by

^Nnyn

h=l

Where : L = The number of strata.

Nh =z The total size (number of units) of stratum

h (Är=l, ...,L).

N = The total number of units in all strata
30 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

If the strata sizes are

I. Pine = 320 acres = Nj

II. Bottom-land hardwoods = 140 acres = Nn

III. Upland hardwoods = 340 acres = Nm

Total = 800 acres = N

Then the estimate of the population mean is

. _ (320) (610) + (140) (737) + (340) (304)

Vat — gQQ

= 502.175 cubic feet per acre

For the estimate of the population total (Yst), simply omit the

divisor N.

Yst = S NKVJ, = 320(610) + 140(737) + 340(304) = 401,740

Alternatively,

f,t = Ny,t = 800 (502.175) =z 401,740

Standard errors.—To determine standard errors, it is first

necessary to obtain the estimated variance among individuals

within each stratum (s^^). These variances are computed in the

same manner as the variance of a simple random sample. Thus,

the variance within Stratum I (Pine) is

(5702 ^ 6402 -I- ... + 7002) _ (61^0)^

Sj2 = i5

(10-1)

_ 3,794,000 - 3,721,000

" 9

= 8111.1111

Similarly,

S//2 = 15,556.6667

s,,,2 ^ 12,204.4444

From these values we find the standard error of the mean of a

stratified random sample (»»,*) by the formula

Where: %» = Number of units observed in stratum h.
ELEMENTARY FOREST SAMPLING 31

This looks rather ferocious and does get to be a fair amount of

work, but it is not too bad if taken step by step. For the timber

cruising example we would have

1 [(320)2(8111.1111)/

8002. L 10

(340)2(12,204.4444)/ 10 \1

10

= V383.920659

= 19.594

As a rough rule we can say that unless a l-in-20 chance has

occurred, the population mean is included in the range

y,t ± 2 (Si,,) = 502.175 ± 2(19.594)

= 463 to 541

If sampling is with replacement or if the sampling fraction

within a particular stratum (njJNn) is small, we can omit the

finite-population correction (l — -rr-) for that particular stratum

when calculating the standard error.

The population total being estimated by í"«* = Nyst, the standard error of fat is simply

Sf^^ - Nsy^^ = 800(19.594) = 15,675

Discussion.—Stratified random sampling offers two primary

advantages over simple random sampling. First, it provides separate estimates of the mean and variance of each stratum. Second,

for a given sampling intensity, it often gives more precise estimates of the population parameters than would a simple random

sample of the same size. For this latter advantage, however, it is

necessary that the strata be set up so that the variability among

unit values within the strata is less than the variability among

units that are not in the same stratum.

Some drawbacks are that each unit in the population must be

assigned to one and only one stratum, that the size of each stratum

must be known, and that a sample must be taken in each stratum.

The most common barrier to the use of stratified random sampling

is lack of knowledge of the strata sizes. If the sampling fractions

are small in each stratum, it is not necessary to know the exact

strata sizes; the population mean and its standard error can be

computed from the relative sizes. If n = the relative size of stratum h, the estimated mean is

L

^, h=l

h=i
32 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

The estimated standard error of the mean is

It is worth repeating that the sizes or relative sizes of the strata

must be known in advance of sampling; the error formulae given

above are not applicable if the observations from which the strata

means are estimated are also used to estimate the strata sizes.

Sample Allocation in Stratified Random Sampling

Assuming we have decided on a total sample size of n observations, how do we know how many of these observations to make

in each stratum? Two common solutions to this problem are known

as proportional and optimum allocation.

Proportional allocation.—In this procedure the proportion of

the sample that is selected in the h*^ stratum is made equal to the

proportion of all units in the population which fall in that stratum.

If a stratum contains half of the units in the population, half of

the sample observations would be made in that stratum. In equation form, if the total number of sample units is to be n, then for

proportional allocation the number to be observed in stratum h is

nn =(t)»

In the previous example, the 30 sample observations were divided

equally among the strata. For proportional allocation we would

have used

nni = (|g) 30 = 12.75 or 13

Optimum allocation.—In optimum allocation the observations

are allocated to the strata so as to give the smallest standard error

possible with a total of n observations. For a sample of size n, the

number of observations (nn) to be made in stratum h under

optimum allocation is

, NnSn

nn={-L
ELEMENTARY FOREST SAMPLING 33

In terms of the previous example the value of N^SK for each

stratum is

Nj8i =320V8111.1111 =320(90.06) =28,819.20

NjiSjj = 140 V15,556.6667 = 140 (124.73) = 17,462.20

NniSnj = 340 V12,204.4444 = 340 (110.47) = 37,559.80

Total = 83,841.20 = S N^Sj,

h=i

Applying these values in the formula, we would get

—(SSS) 30 = 13.4 or 14

Here optimum allocation is not much different from proportional

allocation. Sometimes the difference is great.

Optimum Allocation With Varying Sampling Costs^

Optimum allocation as just described assumes that the sampling

cost per unit is the same in all strata. When sampling costs vary

from one stratum to another, the allocation giving the most information per dollar is

nn:

Where : c» = Cost per sampling unit in stratum h.

The best way to allocate a sample among the various strata depends on the primary objectives of the survey and our information

about the population. One of the two forms of optimum allocation

is preferable if the objective is to get the most precise estimate

of the population mean for a given cost. If we want separate estimates for each stratum and the overall estimate is of secondary

importance, we may want to sample heavily in the strata having

high-value material. Then we would ignore both optimum and proportional allocation and place our observations so as to give the

degree of precision desired for the particular strata.

We cannot, of course, use optimum allocation without having

some idea about the variability within the various strata. The

appropriate measure of variability within the stratum is the

standard deviation (not the standard error), but we need not

know the exact standard deviation (s^) for each stratum. In place

of actual «A values, we can use relative values. In our example, if
34 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

we had known that the standard deviations for the strata were

about in the proportions SiiSn'.Sm = 9:12:11, we could have used

these values and obtained about the same allocation. Where optimum allocation is indicated but nothing is known about the

strata standard deviations, proportional allocation is often very

satisfactory.

Caution! In some situations the optimum allocation formula

will indicate that the number of units (un) to be selected in a

stratum is larger than the stratum (Nn) itself. The common procedure then is to sample all units in the stratum and to recompute

the total sample size (n) needed to obtain the desired precision.

The method of estimating n is discussed in the next section.

Sample Size in Stratified Random Sampling

In order to estimate the total size of sample (n) needed in a

stratified random sample, the following pieces of information are

required :

A statement of the desired size of the standard error of the

mean. This will be symbolized by D.

A reasonably good estimate of the variance (SA^) or standard

deviation (s^) among individuals within each stratum.

The method of sample allocation. If the choice is optimum

allocation with varying sampling costs, the sampling cost

per unit for each stratum must also be known.

Given this hard-to-come-by information, we can estimate the size

of sample (n) with these formulae :

For equal samples in each of the L strata.

NW^Jr ^^nSi?

h=l

For proportional allocation,

h=i

For optimum allocation with equal sampling costs among strata,

V2

h=i
ELEMENTARY FOREST SAMPLING 35

For optimum allocation with varying sampling costs among strata,

When the sampling fractions (TF-) are likely to be very small for

all strata or when sampling will be with replacement, the second

term of the denominators of the above formulae ( S N^s/ J may

be omitted, leaving only NW^.

If the optimum allocation formula indicates a sample (w^)

greater than the total number of units (Nji) in a particular stratum, rih is usually made equal to Nji ; i.e., all units in that particular stratum are observed. The previously estimated sample size

(n) should then be dropped and the total sample size (vf) and

allocation for the remaining strata recomputed omitting the Nn

and SA values for the offending stratum but leaving N and D

unchanged.

As an illustration, assume a population of 4 strata with sizes

(Nji) and estimated variances Sj,^ as follows :

stratum N^ «^2 g^ ¡^^g^ ^fc«h^

1 200 400

2 100 900

3 400 400

4 20 19,600

N = 720 17,800 722,000

With optimum allocation (same sampling cost per unit in all

strata), the number of observations to estimate the population

mean with a standard error of I> = 1 is

The allocation of these observations according to the optimum

formula would be

20 4,000 80,000

30 3,000 90,000

20 8,000 160,000

140 2,800 392,000
36 AGRICULTURE HANDBOOK 232, U.S. ÔEPT. OP AGRICULTURE

The number of units allocated to the fourth stratum is greater

than the total size of the stratum. Thus every unit in this stratum

would be selected (^4 = N^ = 20) and the sample size for the

first three strata recomputed. For these three strata,

S A^ASA = 15,000

S NnSj,^ = 330,000

Hence,

(15,000)^

(7202) (12) -I- 330,000 = 265

And the allocation of these observations among the three strata

would be

'^^(lIS) 265 = 70.7 or 71

«^ = (112) 265 = 53.0 or 53

«8 = (lIS) 265 = 141.3 or 141

Regression Estimators

Regression estimators, like stratification, were developed to increase the precision or efliiciency of a sample by making use of

supplementary information about the population being studied.

If we have exact knowledge of the basal area of a stand of timber,

the relationship between volume and basal area may help us to

iniprove our estimate of stand volume. The sample data provides

mfonnation on the volume-basal area relationship which is then

applied to the known basal area, giving a volume estimate that

may be better or cheaper than would be obtained by sampling

volume alone.

Suppose a 100 percent inventory of a 200-acre pine stand indicates a basal area of 84 square feet per acre in trees 3.6 inches

^^ ¿¿riL^^^ larger. Assume further that on 20 random plots, each

one-fifth acre in size, measurements were made of the basal area

{x) and volume (y) per acre.
ELEMENTARY FOREST SAMPLING 87

Basal area

per acre (a?)

(«9./«.)

88

72

80

96

64

48

76

85

93

110

88

80

Volume per

acre {y)

(Ctt. St.)

1,680

1,460

1,590

1,880

1,240

1,060

1,500

1,620

1,880

2,140

1,840

1,630

Basal area

per acre (x)

iaq. ft.)

82

76

86

73

79

85

84

75

Volume per

acre (y)

(cu. ft.)

1,560

1,560

1,610

1,370

1,490

1,710

1,600

1,440

Total.

Mean.

. 1,620

...81

31,860

1,593

Some values that will be needed later are

n =20

Xy =31,860

y = 1,593

Xy^ =51,822,600

Xxy = 2,635,500

%x =1,620

X =81

Xx^ = 134,210

SSy =Xy^. iXyV — 51,822,600 (31,860)2

20 = 1,069,620

_ SSy _ 1,069,620

(n-1) ~ 19 : 56,295.79

BS^ =Xx^- iXxY 134,210-ilf^ = 2,990

SF^y = Xxy _ (?^)iM ^ 2,635,500 - d^^^O) (31,860) ^ ^^ g^^

N = total number of fifth-acre plots in the population (= 1,000)

The relationship between y and x may take one of several forms,

but here we will assume that it is a straight line. The equation

for the line can be estimated from

yj,=,y+b (X-x)

Where : ya = The mean value of y as estimated from X

(a specified value of the variable X).

y = The sample mean of y (= 1,593).

X = The sample mean of a; (= 81).

b = The linear regression coefficient of y on x.

For the linear regression estimator used here, the value of the

regression coefliicient is estimated by

& =

SPxi 54,840

SSa, ~ 2,990 = 18.34
38 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Thus, the equation would be

'pR = 1,593 + 18.34 (X - 81)

= 107.46 + 18.34 X

To estimate the mean volume per acre for the tract we substitute

for X the known mean basal area per acre.

VR r= 107.46 + 18.34 (84) = 1,648 cubic feet per acre

Standard error.—In computing standard errors for simple random sampling and stratified random sampling, it was first necessary to obtain an estimate (s/) of the variability of individual

values of y about their mean. To obtain the standard error for a

regression estimator, we need an estimate of the variability of

the individual ^/-values about the regression of y onx, A measure

of this variability is the standard deviation from regression

(Sy,a) which is computed by

ocf K^^xy)

kUkioa

{n-2)

1,069,620 (^^'^^^>' 2,990

(20 - 2)

59.53

The symbol Sy,g, bears a strong resemblance to the covariance symbol (Sya,) with which it must not be confused.

Having the standard deviation from regression, the standard

error of yR is

= 13.57

With such a small sampling fraction i-^ = 0.02], the finitepopulation correction ( 1 — ;^) could have been ignored, and the

standard error would be 13.71.

It is interesting to compare s^^ with the standard error that

would have been obtained by estimating the mean volume per acre
ELEMENTARY FOREST SAMPLING 39

from the ¡/-values only. The estimated mean volume per acre would

have been y = 1,593 (compared to 1,648 using the regression

estimator). The standard error of this estimate would be

= 52.52 (compared to a standard error of 13.57

with the regression estimator).

^-^^- '^ (0.98) 20

The family of regression estimators.—The regression procedure

in the above example is valid only if certain conditions are met.

One of these is, of course, that we know the population mean for

the supplementary variable {x). As will be shown in the next

section (Double Sampling), an estimate of the population mean

can often be substituted.

Another condition is that the relationship oi y to x must be

reasonably close to a straight line within the range of x values

for which y will be estimated. If the relationship departs very

greatly from a straight line, our estimate of the mean value of

y will not be reliable. Often a curvilinear function is more

appropriate.

A third condition is that the variance of y about its mean should

be the same at all levels of x. This condition is difficult to evaluate

with the amount of data usually available. Ordinarily the question

is answered from our knowledge of the population or by making

special studies of the variability of y. If we know the way in

which the variance changes with changes in the level of ic a

weighted regression procedure may be used.

Thus, the linear regression estimator that has been described

is just one of a large number of related procedures that enable us

to increase our sampling efficiency by making use of supplementary information about the population. Two other members of this

family are the ratio-of-means estimator and the mean-of-ratios

estimator.

The ratio-of-means estimator is appropriate when the relationship oi y to X is in the form of a straight line passing through

the origin and when the standard deviation of y at any given level

of X is proportional to the square root of x. The ratio estimate

(yn) of mean y is

Where : R — The ratio of means obtained from the sample

X %x

X = The known population mean of x.
40 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

The standard error of this estimate can be reasonably approximated for large samples by

-^/(

g/ + i^V - 2i^g^

')(-F)

Where : z^ = The estimated variance of y.

8/ = The estimated variance of x.

8ay = The estimated covariance of x and y.

It is difficult to say when a sample is large enough for the standard error formula to be reliable, but Cochran (see References,

p. 78) has suggested that n must be greater than 30 and also

large enough so that the ratios Sy/y and s¿/x are both less than 0.1.

To illustrate the computations, assume that for a population of

N = 400 units, the population mean of x is known to be 62 and

that from this population a sample of ?i = 10 units is selected.

The y and x values for these 10 units are found to be

/rvation Vi «t Observation Vt •i

1 8 62 8 .. 11 96

2 13 81 9 .. 5 36

3 5 40 10 .. 12 70

4 6 46 —

5 19 123 Total .. .. 96 680

6 9 74

7 .... 8 52 Mean .. .. 9.6 68

From this sample the ratio-of-means is

9.6 R =

68 0.141

The ratio-of-means estimator is then

yR = RX= 0.141 (62) = 8.742

To compute the standard error of the mean we will need the variances of y and x and also the covariance. These values are computed by the standard formulae for a simple random sample. Thus,

(82 + 132 + ... + 122) _

2 _ 10

(10 - 1)

(622 4. 812 + ... _|. 702) _

2 —

(680)2

10

18.7111

(10 - 1)

(8) (62) + (13) (81) + ... + (12) (70)

= 733.5556

(96) (680)

10

(10 - 1)

= 110.2222
ELEMENTARY FOREST SAMPLING 41

Substituting these values in the formula for the standard error

of the mean gives

^*«=^/P,7111) + (0.1412) (733.5556) - 2(0.141) (110.2222)\

10 )

/ 4ooy

= V-215690

= 0.464

This computation is, of course, for illustrative purposes only. For

the ratio-of-means estimator, a standard error based on less than

30 observations is usually of questionable value.

The mean-of-ratios estimator is appropriate when the relation

0Î y to X is in the form of a straight line passing through the

origin and the standard deviation of 2/ at a given level of x is

proportional to x (rather than to V^) • The ratio (r^) of Vi to Xi

is computed for each pair of sample observations. Then the estimated mean of y for the population is

yR = ÈX

Where : È = the mean of the individual ratios (n), i.e.,

n

n

To compute the standard error of this estimate we must first

obtain a measure (s,.^) of the variability of the individual ratios

(r*) about their mean.

s,^ =

t=l n

{n-D

The standard error for the mean-of-ratios estimator of mean

y is then

-^#(-s)
18 2.00

48 1.98

46 2.35

74 2.32

58 2.17

26 2.23

60 2.05

51 1.92

25 2.16

7 2.00

42 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Suppose that a set of n = 10 observations is taken from a

population OÎN = 100 units having a mean x value of 40 :

Ohaervation Vt *i **4

1 36

2 95

3 108

4 172

5 126

6 58

7 123

8 98

9 54

10 14

Total 21.18

The sample mean-of-ratios is

R = 2L18 ^ 2.118

And this is used to obtain the mean-of-ratios estimator

yß = EX= 2.118 (40) = 84.72

The variance of the individual ratios is

(2.002 + 1.982 + ... + 2.002) _ i^'^-^^)^

S.2 = ^^^-^^ 15_ = 0.022484

Thus, the standard error of the mean-of-ratios estimator is

s,,^40^rö^022484 / iO_\

"" V 10 V 100/

= 1.799

Numerous other forms of ratio estimators are possible, but the

above three are the most common. Less common forms involve

fitting some curvilinear function for the relationship of y to x, or

fitting multiple regressions when information is available on more

than one supplementary variable.

Warning! The forester who is not sure of his knowledge of

regression techniques would do well to seek advice before adapting regression estimators in his sampling. Determination of the
ELEMENTARY FOREST SAMPLING 43

most appropriate form of estimator can be very tricky. The two

ratio estimators are particularly troublesome. They have a simple,

friendly appearance that beguiles samplers into misapplications.

The most common mistake is to use them when the relationship

of 2/ to £c is not actually in the form of a straight line through the

origin (i.e., the ratio oí y to x varies instead of being the same

at all levels of x). To illustrate, suppose that we wish to estimate

the total acreage of farm woodlots in a county. As the total area

in farms can probably be obtained from county records, it might

seem logical to take a sample of farms, obtain the sample ratio

of mean forested acreage per farm to mean total acreage per

farm, and multiply this ratio by the total farm acreage to get the

total area in farm woodlots. This is, of course, the ratio-of-means

estimator, and its use assumes that the ratio of y to x is SL constant (i.e., can be graphically represented by a straight line passing through the origin). It will often be found, however, that the

proportion of a farm that is forested varies with the size of the

farm. Farms on poor land tend to be smaller than farms on fertile

land, and, because the poor land is less suitable for row crops or

nasture. a higher proportion of the small-farm acreage may be

left in forest. The ratio estimate mav be seriously biased.

The total number of diseased seedline-s in a nursery might be

estimated by getting the mean pronortion of infected seedlings

from a number of sample plots and multinlying this nroportion

by the known total number of seedlings in the nursery. Here aerain

we would be assuming that the proportion of infected seedlings

is the same regardless of the number of seedlines per plot. For

many diseases this assumption would not be valid, for the rate

of infection may vary with the seedling density.

Double Sampling

Double sampling was devised to permit the use of regression

estimators when the population mean or total of the supplementary variable is unknown. A large sample is taken in order to

obtain a good estimate of the mean or total for the supplementary

variable (x). On a subsample of the units in this large sample,

the y values are also measured to provide an estimate of the

relationship of y to x. The large sample mean or total of a; is then

applied to the fitted relationship to obtain an estimate of the

population mean or total of y.

Updating a forest inventory is one application of double sampling. Suppose that in 1950 a sample of 200 quarter-acre plots in

an 800-acre forest showed a mean volume of 372 cubic feet per

plot (1,488 cubic feet per acre). A subsample of 40 plots, selected

at random from the 200 plots, was marked for remeasurement m

1955. The relationship of the 1955 volume to the 1950 volume as

determined from the subsample was applied to the 1950 volume

to obtain a regression estimate of the 1955 volume.
44 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

The subsample was as follows :

1955

volume

1950

volume

1955

volume

1950

volume

(l/> (») (V) (x)

370 280 550 430

290 240 550 460

520 410 520 400

490 360 420 390

530 390 490 340

330 220 500 420

310 270 610 470

400 340 460 350

450 360 430 340

430 360 510 380

460 400 450 370

480 380 380 300

430 350 430 290

500 390 460 340

640 480 490 370

660 520 560 440

490 400 580 480

510 430 540 420

270

380

230

270 Total ..18,820 14,790

420 330

530 390 Mean . .470.50 369.75

Xy^ = 9,157,400

5 a;2 zrr 5,661,300

Xxy = 7,186,300

A plotting of the 40 pairs of plot values on coordinate paper

suggested that the variability of y was the same at all levels of x

and that the relationship of t/ to a; was linear. The estimator

selected on the basis of this information was the linear regression

yji¿ = a -}- bX. Values needed to compute the linear-regression

estimate and its standard error were as follows :

Large-sample data (indicated by the subscript 1) :

ni = Number of observations in large sample = 200

N = Number of sample units in population = 3,-200

Xi = Large sample mean of cc == 372 ^^

* Small-sample data (indicated by the subscript 2) :

712 = Number of observations in subsample = 40

^2 = Small sample mean of y = 470.50

ÍC2 = Small sample mean of x = 369.75

SS, =( 212/2

712 )=(»•

157,400 (18,820)2

40 ")- 302,590.0
ELEMENTARY FOREST SAMPLING 45

SS. = (xx^ -^) = (5,661,300 -ílíf^) = 192,697.5

SP^ = [xxy - ^^^) = (7,186,300 - (18,820) a4,790)>j

= 227,605.0

2 _ SS, _ 302,590 rjrr.nrjc.

'y -(n, - 1) - 4Ö^rr - ^'^58.72

The regression coefficient (&) and the squared standard deviation from regression (s^.a,) are ^'

, _ÄP^ _ 227,605.0

^ - ÄS, - 192,697.5 - ^'^^

(SS i^l^\ /" 302 590 0 (227,605.0) n

^ . V^-^-- SS. j (,302,590.0- ^^^^^g^^ j

^^- - (712-2) - 4Ö^='2 -8«8.2617

And the regression equation is

VBd = y2 + b {X — X2)

= 470.50 + 1.18 (X - 369.75)

= 34.2 + 1.18X

Substituting the 1950 mean volume (372 cubic feet) for X gives

the regression estimate of the 1955 volume.

y^a = 34.2 + 1.18(372) = 473.16 cubic feet per plot

Standard error.—The standard error of yRd when the linearregression estimator is used in double sampling is

s- - L .M . ixt-x2y\ /, MTV7Î M

/—"r.lT HI (^^^-OO - 369.75)^\ / '^

= ^888.2617 (^^ + 192,697.5 ) V ~ 200J

7,758.72 7,758.72 /^ 200 \

+ 200 V^ 3,200y

= 7.36 cubic feet

Had the 1955 volume been estimated from the 40 plots without

taking advantage of the relationship of y to x, the estimated mean

would have been

^ 18,820 ^ ^rjQ^Q ç^^jç fgg^ (instead of 473.16)

40
46 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

The standard error of ^ would have been

-NSF^)

_ /7,758.72 /. _ 40 \

- \ 40 V 3,200y

= 13.84 cubic feet (compared to 7.36)

Double sampling with other regression estimators.—^If the

mean-of-ratios estimate is deemed appropriate, the individual

ratios (r< = Vi/Xi) are computed for the n2 observations of the

subsample. The mean of ratios estimate is then

y^a = Rxi

with standard error

Where: Ä =^

""= V--(S){'-S)+ï('-t)

Xri

Xt — Mean x for the large sample of n^ observations

Sr^ z= Variance of r for the subsample

Xr^

%2

"~ ^2 — 1

The ratio-of-means estimate, when appropriate, is

VRd = Rxx

with standard error

s« V('-a(l)'C'^"T''-)+f('-t)

Where : R = y2/x2

Sy^ = Variance of y in the subsample.

s/ = Variance of x in the subsample.

s„- = Covariance of y and x in the subsample.
ELEMENTARY FOREST SAMPLING 47

Sampling When Units are Unequal in Size

(Including PPS Sampling)

Sampling units of unequal size are common in forestry. Plantations, farms, woodlots, counties, and sawmills are just a few of

the natural units that vary in size. Designing and analyzing

surveys involving unequal-sized units can be quite tricky. Two

examples will be used to illustrate the problem and some of the

possible solutions. They also illustrate the very important fact

that no single method is best for all cases and that designing an

efficient survey requires considerable skill and caution.

Example No, 1.—As a first example, suppose that we want to

estimate the mean milling cost per thousand board feet of lumber

at southern pine sawmills in a given area. Available for planning

the survey is a list of the 816 sawmills in the area and the daily

capacity of each. The cost information is to be obtained by personal interview.

In sampling, as in most other endeavors, the simplest approach

that will do the job is the best; complex procedures should be

used only when they offer definite advantages. On this principle

we might first consider taking a simple random sample of the

mills, obtaining the cost per thousand at each, and computing the

arithmetic average of these values. Most foresters would reject

this procedure, and rightly so. The design would give the same

weight to the cost for a mill producing 8,000 feet per day as to

the cost for one cutting 50,000 feet per day. As a result, one thousand feet at the small mill would have a larger representation in

the final average than the same volume at the large mill, and

because cost per thousand is undoubtedly related to mill capacity,

the estimate would be biased.

An alternative that would give more weight to the large mills

would be to take a random sample of the mills, obtain the total

milling cost (i/i) and the total production in MBF (Xi) at each,

and then use the ratio-of-means estimator:

M f -MrT^-ry Total cost at all sampled mills _^

Mean cost per MBi? _ ^^^^j production at all sampled mills " Xx,

This must also be rejected on the grounds of bias. The ratio-ofmeans estimator is unbiased only if the ratio oiytox is the same

at all levels of x. In this example, a constant ratio oí y to x means

that the milling cost per thousand is the same regardless of mill

size—an unlikely situation.

An unbiased procedure and one that would be appropriate in

this situation is sampling with probability proportional to size

(known as pps sampling). The value to be observed on each

sample unit would be the milling cost per thousand board feet of

lumber. Selection of the units with probability proportional to size

is easily accomplished.

First, a list is made of all of the mills along with their daily

capacities and the cumulative sum of capacities.
48 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Mill No.

1 .

2 .

3 .

4 .

814

815

816

DaUy Cumvlative

capacity (MBF) sum

10 10

27 37

8 45

12 57

13

21

11

12,210

12,231

12,242

12,242

Next, numbers varying in size from 1 up to the cumulative sum

for the last mill on the list (12,242) are selected from a table of

random digits. A particular mill is included in the sample when a

number is drawn which is equal to or less than the cumulative

sum for that mill and greater than the cumulative sum for the

preceding mill. Thus, given a random number of 49 we would

select mill number 4 ; for 37 we would select mill number 2 ; for

12,238 we would select mill number 816. An imi)ortant point is

that sampling must be with replacement (i.e., a given mill may

appear in the sample more than once) ; otherwisGj sampling will

not be proportional to size.

After the sample units have been selected and the unit values

(Vi = milling cost per thousand) obtained, the mean cost per

thousand and the standard error of the mean are computed in the

same manner as for simple random sampling with replacement.

Given the following ten observations :

Milling coat Milling coat

Mill per MBF

(dollars)

Mitt per MBF

(doUara) 73 . 12 329 11

541 . 13 804 17

126 . 18 126 18

134 . 14 427 12

423 . 16

703 . 21 Total 152

The estimated mean is

152

y = -j^ = 15.2 dollars per thousand

The standard error of the mean is

TT ^Vi' -

{XViV

n(n—l)

2,408 (152)2

10

10(9)

= 1.04
ELEMENTAKY FOREST SAMPLING 49

Another alternative is to group mills of similar size into strata

and use stratified random sampling. If the cost per thousand is

related to mill size, this procedure may be slightly biased unless

all mills m a given stratum are of the same size. With only a small

withm-stratum spread in mill size, the bias will usually be trivial.

A further refinement would be to group mills of similar size and

use stratified random sampling with pps sampling of units within

strata.

Example No. ^.—-Now, consider the problem of estimating the

total daily production of chippable waste at these mills. Assume

again that we have a list of the mills and their daily capacities.

We might first consider a simple random sample of the mills

with the unit observation being the mean daily production of

chippable waste at the selected mills. The arithmetic average of

these observations multiplied by the total number of mills would

give an estimate of the total daily production of chippable waste

by all mills. This estimate would be completely unbiased. However, because the mills vary greatly in daily capacity and because

total waste production is closely related to total lumber production, there will be a large variation in chippable waste from unit

to unit. This means that the variance among units will be large

and that many observations may be needed to obtain an estimate

of the desired precision. The simple random sample, though unbiased, would probably be rejected because of its low precision.

^ The ratio-of-means estimator is a second alternative. In this design a simple random sample would be selected and for each mill

included in the sample we would observe the mean daily production of chippable waste (Vi) and the mean daily capacity of the

mill in MBF (Xi). The ratio of means

XXi

would give an estimate of the mean waste production per MBF,

and this ratio multiplied by the total capacity of all mills would

estimate the total daily production of chippable waste. It has been

pointed out that the ratio-of-means estimator is unbiased if the

ratio of 2/ to a; is the same at all levels of x. Studies have shown

that although the ratio of waste to lumber production varies with

log size, it is not closely related to mill size—hence the bias, if

any, in the ratio-of-means estimator would be small. Past experience suggests that the variance of the estimate will also be small,

making it preferable to the simple arithmetic average previously

discussed. Note that this is a case where a slightly biased estimator of high precision might be more suitable than an unbiased

estimator of low precision.

Here again, pps sampling would merit consideration. It would

give unbiased estimates of moderately good precision. Stratified
50 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

siönpling with units grouped according to size is another possibility as is the combination of stratification with pps sampling

within strata. Among the acceptable alternatives no blanket recommendation is possible. The best choice depends on many factors,

chief among them being the form and closeness of the relationship

between chippable waste (Vi) and mill capacity (Xi).

Two-Stage Sampling

In some forest sampling, locating and getting to a sampling unit

is expensive, while measurement of the unit is relatively cheap.

It seems logical in these circumstances to make measurements on

two or three units at or near each location. This is called two-stage

sampling, the first stage being the selection of locations, and the

second stage being the selection of units at these locations. The

advantage of two-stage sampling is that it may yield estimates of

a given precision at a cost lower than that of a completely random

sample.

To illustrate the situation and the methods, consider a landowner whose 60,000 acres of timberland are subdivided into

square blocks of 40 acres with permanent markers at the four

corners of each block. A sample survey is to be made of the tract

in order to estimate the mean sawtimber volume per acre. Sample

units will be square quarter-acre plots. These plots will be located

on the ground by measurements made with reference to one of

the corners of the 40-acre blocks.

Travel and surveying time to a block comer are quite high,

hence it seems logical, once the block comer is located, to find and

measure several plots in that block. Thus, the sampling scheme

would consist of making a random selection of n blocks and then

randomly selecting m plots within each of the selected blocks. In

sampling language, the 40-acre blocks would be called primary

sampling units (primaries) and the quarter-acre plots secondary

sampling units (secondaries).

If 2/<y designates the volume of the /*»» sampled plot (/ = 1... m)

on the i^ sampled block, the estimated mean volume per plot (symbolized in two-stage sampling by y) is

i:s Vu

y = '=' '=' mn

The standard error of the estimated mean is

Si = Vir[-=(^-F) + ^(i-f)]

Where : n = Number of primaries sampled.

N = Total number of primaries in the population.

m — Number of secondaries sampled in each of the primaries selected for sampling.
ELEMENTARY FOREST SAMPLING 51

M = Total number of secondaries in each primary.

SB^ = Sample variance between primaries when sampled by

m secondaries per primary (computation procedure

given below).

Sw^ = Sample variance among secondaries within primaries

(computation procedure given below).

The terms SB^ and Sw^ are computed from the equations

t-l\i=l / _ V=l 3=1 /

„ 2 _ V^ W^ OB in- 1)

t=iV=i /

n{m — 1)

Since Vij is the observed value of a secondary unit, S Vu is the

3=1

total of all secondary units observed in the i^ primary (or the

n m

primary total), and S S Va is the grand total of all sampled

t=l 3=1

secondaries. Hence, the above equations, expressed in words, are

ls2) \ /[s(Secondaries)]^ \

ies / I Total no. of J

lary/ \secondaries sampled/

Sw^ =

in

I 2 (Primary totals^)

I No. of secondaries

\sampled per primary/

(n - 1)

/ ¿ (Primary totals^) \

S (Secondaries^) - I NO. of secondaries )

\ sampled per primary /

n{m — 1)

Readers familiar with analysis of variance procedures will recognize SB^ and Sw^ as the mean square between and within primaries

respectively.

The computations are not so difficult as the notation might suggest. Suppose we had sampled m = 3 quarter-acre plots (secondaries) within each of w = 4 blocks (primaries) and obtained the

following data:
52 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Block Plot 5ecafi<{ari/ Prima¿ry

(secondary) voltte« totola

rctt&io feet) (cubic feet)

1 1 147

2 180

3 206 533

2 1 312

2 265

3 300 877

3 1 220

2 280

3 210 710

4 1 250

2 232

3 185 667

2,787 2,787

The estimated mean per plot is

^ ^ (147 + 180 + ... + 185) ^ 2^^ 232 25 ^„^.^ ^^^ ^^^ p,^^

To get the standard error of # we first compute SB^ and STT^.

(533^ -}-...-f 667^) (2,787)^

,._ 3 (3) (4)

'' - (4-1)

667,402.3333 - 647,280.7500

Sw'

3

= 6,707.1944

(533^ + ... + 667^)

3

(1472 +1802+ ...+1852y

(4) (3-1)

675,463.0000 - 667,402.3333

~ 8

= 1,007.5833

Since the total number of 40-acre blocks in the 60,000 acres is

N = 1,500 and the total number of quarter-acre plots in each

40-acre block is ilf = 160, the estimated standard error of the

mean is

= ^/(3^4)[6.70'^•l«44(l-ïJ5ö)

+ (4)

4

(1,007.5833)/ 3 X]

1,500 V 160/J

^ [6,689.3085 + 2.6365]

= 23.61
ELEMENTARY FOREST SAMPLING 53

The estimated mean per plot is 232.25 cubic feet. The standard

error of this estimate is 23.61 cubic feet. As the plots are onequarter acre in size, the estimated mean volume per acre is 4 (#)

= 929 cubic feet. The standard error of the mean volume per acre

is 4(sj) = 94.44.

An estimate of the total volume and its standard error can be

obtained either from the mean per plot or mean per acre volumes

and their standard errors. The mean per plot is 232.25 it 23.61.

To expand this to the total, each figure must be multiplied by the

number of quarter-acre plots in the entire tract (= 240,000) ; the

estimated total is

55,740,000 ± 5,666,400.

The mean per acre is 929 ± 94.44. To expand this, each figure

must be multiplied by the total number of acres in the tract

(= 60,000). Thus, the estimated total is

55,740,000 ± 5,666,400 as before.

Small sampling fractions.—If the number of primary units

sampled {n) is a small fraction of the total number of primary

units (N)y the standard error formula simplifies to

mn

This reduced formula is usually applied where the ratio n/N is

less than 0.01. In the example above, the sampling fraction for

primaries was 4/1,500, so we could very well have used the

short formula. The estimated standard error would have been

-4MOT^^V558:9329

= 23.64 (instead of 23.61 by the longer formula).

When n/N is fairly large but the number of secondaries (m)

sampled in each selected primary is only a small fraction of the

total number of secondaries (M) in each primary, the standard

error formula would be

'.=^/i[•. (-^)+=r]

Sample size for two-stage sampling,—For a fixed number of

sample observations, two-stage sampling is usually less precise

than simple random sampling. The advantage of the method is

that by reducing the cost per observation it permits us to obtain

the desired precision at a lower cost. . jUsually the precision and cost both increase as the number ot

primaries is increased and the number of secondaries (m) per

sampled primary is decreased. The cost may be reduced by taking
54 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

fewer primaries and more secondaries per primary, but precision

usually suffers. This suggests that there is a number (m) of secondaries per primary that will be optimum from the standpoint

of giving the greatest precision for a given amount of money. The

value of m that is optimum depends on the nature of the population variability between primaries and among secondaries within

primaries, and on the relationship between the cost per primary

and the added cost per secondary.

The population variability between primaries is symbolized by

Ö/2 and the variability within primaries by on^. Note that these

are population values, not sample values. Occasionally we will

have some Imowledge of ar^ and ou^ from previous work with the

population. More often, it will be necessary to take a preliminary

sample to estimate the population variabilities. From this presample, we compute SB^ and Sw^ according to the formulae in the

discussion of the error of a two-stage sample. Then our estimates

of the population variability within and between primaries are

SB^ - Sir"

lit/

The cost of locating and establishing a primary unit (not counting overhead costs) is symbolized by Cp. The additional cost of

getting to and measuring a secondary unit after the primary has

been located is symbolized by CsGiven the necessary cost and variance information, we can

estimate the optimum size of m (say rrio) by

-mm

If mo is greater than the number of secondaries per primary (M),

the formula value is ignored and m«, is set equal to M.

Once rrio has been estimated, the number of primary units (with

rrio secondaries per primary) needed to estimate the mean with a

specified standard error (D) is

(•^^^)

"'+^"'+•¿0

Where : N = Total number of primaries in the population.

M = Total number of secondaries per primary.

Numerical example.—Suppose that we wish to estimate a population mean with a standard error of 10 percent or less. We have

defined the population as being composed of N = 1,000 primaries

with M = 100 secondaries per primary.
ELEMENTARY FOREST SAMPLING 55

As we know nothing of the variability between or within these

primaries nor about the costs, we take a preliminary sample

consisting of eight primaries with two secondaries per primary.

Results are as follows:

Data from preliminary survey

Observed values Primary

Primary of secondaries total

1 34 42 76

2 36 17 53

3 41 56 97

4 62 40 102

5 82 94 176

6 16 38 54

7 22 41 63

8 93 50 143

Total = 764

From this preliminary sample, we compute

V = 981.8571

Sw^ = 248.2500

. = ^ = 47.75

Therefore, the estimates of the population variances between and

within primaries are

0„2 _ s^2 _ 248.25

3 _ sl^^^ _ 981.8571 - 248.2500 ^ ^

"^^ - m ~~ 2

Assume also that the preliminary sample yields the following cost

estimates :

Cp = $14.00

c, = $ 1.20

Then our estimate of the optimum number of secondaries to be

observed in each primary is

m<

__ //248.25 \ /14.00\

-\\366.8036 y V 1.20/

= V (.6768) (11.6667)

= V7.8960

= 2.8
56 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

Since we can't observe a fraction of a unit, we must now decide

whether to take two or three secondaries per primary. To do this,

we estimate the number of primaries needed for an m of 2 and for

an m of 3, compute the cost of the two alternatives and select the

less expensive one.

Our preliminary sample gave an estimate of the mean of 47.75

and, since we have specified a standard error of 10 percent, this

means we want D = (0.10) (47.75) = 4.775 or 4.8.

If m = 2, the number of primaries needed for the desired precision would be

_ 490.9286

~ 23.4093

= 20.97

or, n = 21

There will be 21 primaries at a cost of $14 each and 2(21) = 42

secondaries at a cost of $1.20 each, so that the total survey cost

(exclusive of overhead) will be $344.40.

If m = 3, the number of primaries will be

248.25 >

(366.8036+2á|?5)

(4.8)^+^(366.8036+^)

449.5536 ^ ^^^^0

~" 23.4093

or, n = 20

The cost of this survey will be

20(14.00) +60(1.20) =352.00

As the first alternative gives the desired precision at a lower cost,

we would sample n = 21 primaries and m = 2 secondaries per

primary.

Systematic arrangement of secondaries.—Though the potential

economy of two-stage sampling has been apparent and appealing
ELEMENTARY FOREST SAMPLING 57

to foresters, they have displayed a reluctance to select secondary

units at random. Primary sampling points may be selected at random, but at each point the secondaries will often be arranged in

a set pattern. This is not two-stage sampling in the sense that

we have been using the term, though it may result in similar increases in sampling efficiency. It might be called cluster sampling,

the cluster being the group of secondaries at each location. The

unit of observation then is not the individual plot but the entire

cluster. The unit value is the mean or total for the cluster. Estimates and their errors are computed by the formulae that apply

to the method of selecting the cluster locations.

Within each primary the clusters should be selected so that

every secondary has a chance of appearing in the sample. If certain portions of the primaries are systematically excluded, bias

may result.

Two-Stage Sampling With Unequal-Sized Primaries

The two-stage method of the previous chapter gives the same

weight to all primaries. This hardly seems logical if the primaries

vary greatly in size. It would, for example, give the same weight

to a 10,000-acre tract as to a 40-acre tract. There are several modified methods of two-stage sampling which take primary size into

account.

Stratified two-stage sampling.—One approach is to group equalsized primaries into strata and apply the standard two-stage methods and computations within each stratum. Population estimates

are made by combining the individual stratum estimates according to the stratified sampling formulae. This is a very good design

if the size of each primary is known and the number of strata is

not too large. If the number of primaries is small, it may even be

feasible to regard each primary as a stratum and use regular

single-stage stratified sampling.

Selecting primaries with probability proportional to size.—Another possibility is to select primaries with probability proportional to size (pps) and secondaries within primaries with equal

probability. Selection of primaries must be with replacement, but

secondaries can be selected without replacement. A new set of

secondaries should be drawn each time that a given primary is

selected so that a secondary that was selected during one sampling

may again be selected during some subsequent sampling of that

primary.

After the observations have been made, the sample mean (Vi)

is computed for each of the n primaries included in the sample.

These primary means are then used to compute an estimate of the

population mean by
58 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

The standard error of the mean is

n(n — 1)

If only one secondary is selected in each selected primary, this

procedure becomes identical to simple random sampling.

If there is any relationship between the primary size and its

mean, pps sampling may give estimates of low precision. The precision can be improved by combining stratified two-stage sampling

and pps selection of primaries. Primaries of similar size are

grouped into strata and within each stratum selection of primaries

is made with probability proportional to size. Strata means and

variance are computed by the formulae for two-stage sampling

with pps selection of primaries.

Selection of primaries with equal probability.—The procedures

that have been discussed so far require reasonably accurate information about the size of each primary in the population—information that is often lacking. An alternative technique requires

knowledge only of the size of the primaries actually included in

the sample and of the total number of primaries in the population.

The method involves selection of n primaries and mi secondaries

within the i^ selected primary. At each level, sampling is with

equal probability and without replacement. The number of secondaries sampled (mi) may vary or remain constant. The sample

primary mean (yi) is computed for each selected primary and

from these the population mean is estimated as

. t=i

Ci^')

where : n = Number of primaries sampled.

yt = Mean per secondary in the i^ sampled primary.

Mi = Total number of secondary units in the t**» sampled

primary (this can be an actual or a relative measure

of size).

The standard error of this estimate is

^^-yy¡n-i\{%M,y "*"(5!r<)2 - JXMdJxTd) V~"ñ)

where : n = Number of primaries sampled.

N = Total number of primaries.

T, = (Mm)
ELEMENTARY FOREST SAMPLING 59

For an illustration of the computations, suppose that we wished

to estimate the mean board-foot volume on a population of 426

woodlots. Four woodlots (primaries) are selected at random, and

withm each woodlot the board-foot volume is measured on two

randomly selected one-fifth-acre plots. For each woodlot selected,

the acreage is also determined. Since one-fifth-acre plots were

used, the value of M< for the i*^ woodlot will be 5 times its acreage.

Assume the observed values are as follows :

Primary

SampUd meana Woodlot

woodlot pi^,^ values (Vi) acreage M^ Mi^i - T^

Bd. ft. Bd. ft.

1 620 740 680 110 550 374,000

2 585 475 530 26 130 68,900

3 590 730 660 54 270 178,200

4 960 820 890 60 300 267,000

1,250 888,100

Then

^ "^ ^{%MÍ = ^^1250 == ^^^'^^^^^^^^^^*P^^fifth-acre plot.

The values needed to compute the standard error are

XMi^ r= 482,300 %T^ = 247,667,450,000

XMiTi = 342,871,000

(5M<)2 = 1,562,500 i^Ti)^ = 788,721,610,000

(XMi) (%Ti) = 1,110,125,000

Hence,

<î._7in48 //4Y 482,300 , 247,667,450,000

Sy - au.48 VV3Al,562,500 "^ 788,721,610,000

(2) (342,871,000) / fpc \

1,110,125,000 Vignoredy

= 710.48 V0.00662295

= 57.82 board feet.

This estimate of the mean will be slightly biased if there is any

relationship between the primary size and the mean per unit in

that primary. The bias is generally not serious for large samples

(more than 30 primaries).

An unbiased equaUvrohahility estimator.—If the bias incurred

by use of the above estimator is expected to be large, an unbiased

estimate can be obtained. In addition to the information required

for the biased procedure, we must also know the total number of

secondaries (M) in the population. As in the case of the biased

estimator, n primaries are selected with equal probability and

within each primary mi secondaries are observed. The mean per
60 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

unit (Vi) is computed for each primary and used to estimate the

population mean

The standard error of the mean is

c -K /^ '^'^ n (. n\

^'~ M\ n(n-l) V^~ Ñ)

Now, assume that the 426 woodlots of the previous example

have a total area of 26,412 acres. Then, because the secondary

units are one-fifth acre in size, the total number of secondaries in

the population is M = 132,060. With the same sample data the

unbiased estimate of the population mean per unit would be

^ = 4(isloeo) (888,100) z= 716.21 board feet per plot.

The standard error is

426 / (^'^^'ÖÖÖ' + . . • + 267,0002) - (888400)^

132,060 \ 4(3)

/ fpc \

\ignored/

= 0.003226 V4,207,253,958

= 209.25 board feet.

The standard error of the unbiased estimate (209.25) as compared to that of the biased estimate (57.82) shows why the latter

is often preferred. But, if the size of all primaries is known, the

bias of the biased estimator can be reduced and the precision of

the unbiased estimator increased by grouping similar sized primaries and using these estimating procedures in conjunction with

stratified sampling.

Systematic Sampling

As the name implies, and as most foresters know, the units included in a systematic sample are selected not at random but

according to a pre-specified pattern. Usually the only element of

randomization is in the selection of the starting point of the pattern, and even that is often ignored. The most common pattern

is a grid having the sample units in equally spaced rows with a

constant distance between units within rows.
ELEMENTARY FOREST SAMPLING 61

To the disdain of some statisticians, the vast majority of forest

surveys have been made by some form of systematic sampling.

There are two reasons: (1) the location of sample units in the

field is often easier and cheaper, and (2) there is a feeling that

a sample deliberately spread over the entire population will be

more representative than a random sample.

Statisticians usually will not argue against the first reason.

They are less willing to accept the second. They admit the possibility, sometimes even the probability, that a systematic sample

will give a more precise estimate of the true population mean

(i.e., be more representative) than would a random sample of the

same size. They point out, however, that estimation of the sampling error of a systematic survey requires more knowledge about

the population than is usually available, with the result that the

sampler can seldom be sure just how precise his estimate is. The

common procedure is to use random sampling formulae to compute the errors of a systematic survey. Depending on the degree

and the way in which the population falls into patterns, the precision may be either much lower or much higher than that suggested by the random formulae. If there is no definite pattern in

the unit values in the population, the random formulae may give

a fair indication of the sampling precision. The difficulty is in

knowing which condition applies to a particular sample.

The well-known procedure of superimposing two or more systematic grids, each with randomly located starting points, does

provide some of the advantages of systematic sampling along with

a valid estimate of the sampling error. In this procedure each grid

becomes, in effect, a single observation and the error is estimated

from the variability among grids. Locating plots in the field becomes more difficult as the number of grids increases, however,

and it would seem as though the advantage of representativeness

could be obtained more easily and efficiently by stratified sampling

with small blocks serving as strata.

Despite the known hazards, foresters are not likely to give up

systematic sampling. They will usually take the precaution of

running the lines of plots at right angles rather than parallel to

ridges and streams. In most cases, sampling errors will be computed by formulae appropriate to random sampling. Experience

suggests that a few of these surveys will be very misleading, but

that most of them will give estimates having precision as good

as or slightly better than that shown by the random sampling

formulae. Some statisticians will continue to bemoan the practice

and a few of them will keep searching for a workable general solution to the problem of error estimates (though at least one very

eminent statistician doubts that a workable solution exists).

SAMPLING METHODS FOR DISCRETE VARIABLES

Simple Random Sampling—Classification Data

Assume that from a large batch of seed 50 have been selected

at random in order to estimate the proportion (p) that are sound.
62 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

Assume also that cutting or hammering discloses that 39 of the 50

seeds were sound. Then our estimate (p) of the proportion that

is sound is

_ _ Number having the specified attribute

^ ~ Number observed

39

- 50

=: 0.78

Standard error of estimate.—The estimated standard error of

p is

_ ¡pq-P) I. ^

where : n =r number of units observed.

In this example N is extremely large relative to n, and so the

finite-population correction could be ignored

Sp = Y

(0.78) (1 - 0.78)

(50 - 1)

= 0.05918

Confidence limits.—For certain sample sizes (among them, n

= 50), confidence limits can be obtained from table 3, page 87.

In this example we found that in a sample of ri = 50 seeds, 39

were sound. The estimated proportion sound was 0.78 and, as

shown in table 3, the 95-percent confidence limits would be 0.64

and 0.88. For samples of 100 and larger the table does not show

the confidence limits for proportions higher than 0.50. These can

easily be obtained, however, by working with the proportion of

units not having the specified attribute. Thus suppose that, in a

sample oin = 1,000 seeds, 78 percent were sound. This is equivalent to saying that 22 percent were not sound, and the table shows

that for n = 1,000 the 95-percent confidence interval for an observed fraction of 0.22 is 0.19 to 0.25. If the true population proportion of unsound seed is within the limits 0.19 and 0.25, the

population proportion of sound seed must be within the limits 0.75

and 0.81.

Confidence intervals for large samples.—For large samples, the

95-percent confidence interval can be computed as

[^» + -t]

Assume that a sample ofn = 250 units has been selected and that

70 of these units are found to have some specified attribute. Then,
ELEMENTARY FOREST SAMPLING 63

And,

1(0 28) (0 72) Sp = J— 249 (ignoring the finite-population correction)

= 0.02845

Then, the 95-percent confidence interval

= 0.28 zh [2(0.02845) +2¿o)]

= 0.28 ± 0.059

= 0.221 to 0.339

Thus, unless a l-in-20 chance has occurred, the true proportion

is somewhere within the limits 0.22 and 0.34. For a 99-percent

confidence interval we would multiply Sp by 2.6 instead of 2. (For

samples of n = 250 or 1,000, the confidence interval could, of

course, be obtained from table 3. For this example the table gives

0.22 to 0.34 as the limits.)

The above equation gives what is known as the normal approximation to the confidence limits. As noted, it can be used for large

samples. What qualifies as a large sample depends on the proportion of items having the specified characteristic. As a rough guide,

the normal approximation will be good if the common logarithm

of the sample size (n) is equal to or greater than

1.5-|-3(|P-0.5|)

where : P = our best estimate of the true proportion of the population having the specified attribute.

|P — 0.5| = the absolute value (i.e., algebraic sign ignored) of

the departure of P from 0.5.

Thus, if our estimate of P is 0.20 then \P — 0.5| is equal to 0.3

and, if we are to use the normal approximation, the log of our

sample size should be greater than

1.5 + 3(0.3) =2.4

Or n must be over 251 (2.4 = log 251).

Sample size.—^Table 3 may also be used as a guide to the number

of units that should be observed in a simple random sample to

estimate a proportion with a specified precision. Suppose that we

are sampling a population in which about 40 percent of the units

have a certain attribute and we wish to estimate this proportion

to within ±: 0.15 (at the 95-percent level). The table shows that

for a sample of size 30 having p = 0.4 the confidence limits would

be 0.23 and 0.60. Since the upper limit is not within 0.15 of

p = 0.4, a sample of size 30 would not give the necessary precision. A sample of % = 50 would give limits of 0.27 and 0.55. As

each of these is within 0.15 or ^ = 0.4, we conclude that a sample

of size 50 would be adequate.
64 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

If the table suggests that a sample of over 100 will be needed,

the size can be estimated by

n = r- for 95-percent confidence E ^1

(4)(P)(1-P) ' N

1

E^ _4_1.

\ "f" XT

for 99-percent confidence

(6.76) (P) (1 - P) ' N

where : E = The precision with which P is to be estimated.

N = Total number of units in the population.

The table indicates that to estimate a P of about 0.4 to within

E = ±1 0.05 (at the 95-percent confidence level) would require

somewhere between 250 and 1,000 observations. Using the first of

the above formulae (and assuming N = 5,000) we would find,

T- = 357 (0.05)2

(4) (0.4) (0.6) ' 5,000

If we have no idea of the value of P, we will have to make a

guess at it in order to estimate the sample size. The safest course

is to guess a P as close to 0.5 as it might reasonably occur.

How to select a seed at random.—If we were trying to estimate

the proportion of trees in a stand having a certain disease, it

would be difficult to select the individual trees at random and then

locate them in the field for observation. In some populations, however, the individuals themselves are randomly located or can easily

be made so. A batch of seed is such a population. By thoroughly

mixing the seed prior to sampling, it is possible to select a number of individuals from one position in the batch and assume that

this is equivalent to a completely random sample. Those who have

sampled seed warn against mixing in such a manner that the light

empty seeds tend to work together towards the top of the pile.

The sample could be taken with a small scoop or a seed probe

which picks up approximately the number of seed to be examined.

As a precaution, most seed samplers will use a scoop that selects

only a fraction of the desired number of seeds and will take

samples from several places in the pile and combine them.

Cluster Sampling for Attributes

In attribute sampling the cost of selecting and locating an individual is usually very high relative to the cost of determining

whether or not the individual has a certain characteristic. Because

of this, some form of cluster sampling is usually preferred over

simple random sampling. In cluster sampling, a group of individuals becomes the unit of observation, and the unit value is the

proportion of the individuals in the group having the specified

attribute.

In estimating the survival percent of a plantation it would be

possible to choose individual trees for observation by randomly
ELEMENTARY FOREST SAMPLING 65

selecting pairs of numbers and letting the first number stand for

a row and the second number designate the tree within that row.

But it would obviously be inefficient to ignore all of the trees that

must be passed to get to the one selected. Instead, we would probably make survival counts in a number of randomly selected rows

and (assuming the same number of trees were planted in each

row) average these to estimate the survival percent. This is a

form of cluster sampling, the cluster being a row of planted trees.

The germination percent of a batch of seed might also be estimated by cluster sampling. Here the advantage of clusters comes

not in the selection of individuals for observation but from avoiding some hazards of germination tests. Such tests are commonly

made in small covered dishes. If all the seeds are in a single dish,

any mishaps (e.g., overwatering or fungus attack) may affect

the entire test. To avoid this hazard, it is common to place a fixed

number of seeds (one or two hundred) in each of several dishes.

The individual dish then becomes the unit of observation and the

unit value is the germination percent for the dish.

When clusters are fairly large and all of the same size, the procedures for computing estimates of means and standard errors

are much the same as those described for measurement data. To

illustrate, assume that 8 samples of 100 seeds each have been

selected from a thoroughly mixed batch. The 100-seed samples are

placed in 8 separate germination dishes. After 30 days, the following germination percentages are recorded :

Dish No. 1123456781 Total

Germination (pet.) | 84 88 86 76 81 80 85 84 | 664

If Vi is the germination percent in the ¿*^ dish, the mean germination percent would be estimated by

=1

n ~ %

The variance of v would be computed by

(g^^y ,.., . oo. . , o..^ (664)2 ¿y,2_V=i / (842 + 882 + . . . + 842)

. 2

1=1

(^ - 1) ~ 7

= 14.5714

Whence the standard error of p can be obtained as

_ /14.5714 _ ^ gg (ignoring the finite-population correction)
66 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

Note that, in cluster sampling, n stands for the number of clusters

sampled and N is the number of possible clusters in the population.

As in simple random sampling of measurement data, a confidence interval for the estimated percentage can be computed by

Student's t

95-percent confidence interval = p ±t (s¿)

Where: t = Value of Student's t at the 0.05 level with n — 1

degrees of freedom. Thus, in this example, t would have 7 degrees

of freedom and í 05 would be 2.365. The 95-percent confidence

interval would be

83.0 ± (2.365) (1.35) = 83.0 ±: 3.19

= 79.8 to 86.2

Transformation of percentages.—If clusters are small (less

than 100 units per cluster) or if some of the observed percentages are greater than 80 or less than 20, it may be desirable to

transform the percentages before computing means and confidence

intervals. The common transformation is arcsin VPercent. Table

4, page 89, makes it easy to transform the observed percentages.

For the data in the previous example, the transformed values

would be

Diah No.

1 ..

2 ..

3 ..

4 ..

5 ..

84

88

86

76

81

66.4

69.7

68.0

60.7

64.2

80

85

84

63.4

67.2

66.4

Total 526.0

The mean of the transformed values is

526.0

8

= 65.75

The variance of these values is

(66.42 + . • . + 66.4^)

(526)2

= 8.1486

And the standard error of the mean transformed value is

s¿ = yj^^^^ = V1.0186 = 1.009

So the 95-percent confidence limits would be (using ios for 7 df's

= 2.365)

CI = 65.75 ± (2.365) (1.009) =: 65.75 ± 2.39

= 63.36 to 68.14
ELEMENTARY FOREST SAMPLING 67

Referring to the table again, we see that the mean of 65.75

corresponds to a percentage of 83.1. The confidence limits correspond to percentages of 79.9 and 86.1. In this case the transformation made little difference in the mean or the confidence

limits, but in general it is safer to use the transformed values

even though some extra work is involved.

Other clvster-sampling designs.—If we regard the observed or

transformed percentages as equivalent to measurements, it is easy

to see that any of the designs described for continuous variables

can also be used for cluster sampling of attributes. In place of

individuals, the clusters become the units of which the population

is composed.

Stratified random sampling might be applied when we wish to

estimate the mean germination percent of a seed lot made up of

seed from several sources. The sources become the strata, each

of which is sampled by two or more randomly selected clusters

of 100 or 200 seeds.

With seed stored in a number of canisters of 100 pounds each,

we might use two-stage sampling, the canisters being primary

sampling units and clusters of 100 seeds being secondaries. If the

canisters differed in volume, we might sample canisters with

probability proportional to size.

Cluster Sampling for Attributes—Unequal-Sized Clusters

Frequently when sampling for attributes, we find it convenient

to let a plot be the sampling unit. On each plot we will count the

total number of individuals and the number having the specified

attributes. Even though the plots are of equal area, the total number of individuals may vary from plot to plot; thus, the clusters

will be of unequal size. In estimating the proportion of individuals

having the attribute, we probably would not want to average the

proportions for all plots because that would give the same weight

to plots with few individuals as to those with many.

In such situations, we might use the ratio-of-means estimator.

Suppose that 2,4,5-T has been sprayed on an area of small scrub

oaks and we wish to determine the percentage of trees killed. To

make this estimate, the total number of trees (Xi) and the number

of dead trees (Vi) is determined on 20 one-tenth-acre plots.

Plot No. of No. of dead Plot No. of No. of dead

trees (x^) trees T»,; trees (x^) trees (y^)

1 15 11 13 26 16

2 42 32 14 160 126

3 .... ... 128 98 15 103 80

4 86 42 16 80 58

5 97 62 17 32 25

6 8 6 18 56 44

7 28 22 19 49 24

8 .... 65

71

51

48

20 84 59

9 ....

10 .... ... . 110 66 Total . . 1,351 960

11 63 58

12 .... 48 32 Mean . 67.55 48.0
68 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

The ratio-of-means estimate of the proportion of trees killed is

The estimated standard error of p is

Where : Sy^ = Variance of individual y values.

Sa,2 _: Variance of individual x values.

Sy:j, = Covariance of y and x,

n = Number of plots observed.

In this example

. 2 .

(IP + 322 + . . . + 592) _^

JÖ — = 892.6316

1,3512 (152 -f 422 + . . . + 842) _

-?^ = 1,542.4711 19

(960) (1,351) (11) (15) + (32) (42) + ... + (59) (84) --

20

19

= 1,132.6316

With these values (but ignoring the fpc),

(67.55)2

= 0.026

892.6316 + (0.7106)2 (1,542.4711)

-2(0.7106) (1,132.6316)

20

As in any use of the ratio-of-means estimator, the results may

be biased if the proportion of units in a cluster having a specified

attribute is related to the size of the cluster. For large samples,

the bias will often be trivial.

Sampling of Count Variables

Statistical complications often arise in handling data such as

number of weevils in a cone, number of seedlings on a one-tenthmilacre plot, and similar count variables having no fixed upper

limit. Small counts and those with numerous zeroes are especially

troublesome. They tend to follow distributions (Poisson, negative

binomial, etc.) that are diflScult to work with. If count variables

cannot be avoided, the amateur sampler's best course may be to
ELEMENTARY FOREST SAMPLING 69

define the sample units so that most of the counts are large and

to take samples of 30 units or more. It may then be possible to

apply the procedures given for continuous variables.

In order to estimate the number of larvae of a certain insect

in the litter of a forest tract, one-foot-square litter samples were

taken at 600 randomly selected points. The litter was carefully

examined and the number of larvae recorded for each sample. The

counts varied from 0 to 6 larvae per plot. The number of plots on

which the various counts were observed were

Count =0 1 23456 Total

Number

of plots = 256 224 92 21 4 1 2 600

The counts are very close to a Poisson distribution (see page

6). To permit the applications of normal distribution methods,

the units were redefined. The new units were to consist of 15 of

the original units selected at random from the 600. There were

to be a total of 40 of the new units, and unit values were to be

the total larvae count for the 15 selected observations. The values

for the 40 redefined units were

14 13 16 13 13 14 15 12

16 18 11 7 9 10 11 10

12 14 13 14 14 13 9 17

15 8 12 5 13 15 13 10

12 12 20 10 9 14 15 13

Total = 504

By the procedures for simple random sampling of a continuous

variable, the estimated mean (y) per unit is

The variance (Sy^) is

(504)2 (142 4-162 + .. . + 132)- 40

Oy — gg

= 8.8615

With correction for finite population ignored, the standard error

of the mean {Sy) is

^F8615

40

= 0.47
70 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

The new units have a total area of 15 square feet; hence to

estimate the mean number of larvae per acre the mean per unit

must be multiplied by

^?f«= 2,904

Thus, the mean per acre is

(2,904) (12.6) = 36,590.4

The standard error of the mean per acre is

(2,904) (0.47) = 1,364.88

As an approximation we can say that unless a l-in-20 chance

has occurred in sampling, the mean count per acre is within the

limits

36,590.4 ± 2(1,364.88)

or,

33,860 to 39,320

SOME OTHER ASPECTS OF SAMPLING

Size and Shape of Sampling Units

The size and shape of the sampling unit may profoundly affect

the cost of the survey, its precision, or both. No attempt will be

made here to oifer an exhaustive study, but an example may illustrate the problem and a general approach to its solution.

Consider a preharvest inventory in a nursery containing 1,000

beds of slash pine, each bed 500 feet long and 4 feet wide. Conventional practice in this nursery has been to sample the beds by

observing the total number of plantable seedlings in a 1- by 4-foot

sampling frame laid crosswise at five randomly chosen locations

in each bed. The process is laborious and time consuming, totaling

5,000 observations, or nearly a mile of bed. The nurseryman would

like to know if a frame 6 inches wide would be better than the

conventional 12-inch frame.

One practical way to judge among sampling units is to compare

the total cost of surveys made with each unit, with the restriction

that both methods shall afford equal precision. For example,' if

the cost per observation with the 6-inch frame is ííi, then for Ui

observations the cost of the survey (exclusive of overhead costs,

which are assumed to be the same for both size units) is

Similarly, for the 12-inch frame, we can say

C2 = 112^2

3 For illustrative purposes the nursery survey will be treated as a simple

random sample, thougrh the specification of a set number of plots in each bed

makes it a stratified design.
ELEMENTARY FOREST SAMPLING 71

Then the cost of the 6-inch frame relative to the cost of the 12-

inch frame is

Ci fiidi

If estimates of population variance Si^ and «2^ are available, variance of the population totals (ignoring the fpc) may be written

and

"'-m

m

where : Ni and N2 = Number of units of each size in the population. Now if the two methods are to give equal precision for the

estimate of total production,

ST.'

or,

and, solving for n2

« (?)=* (Ï)

This last quantity may be simplified by remembering that the total

number of 6-inch units (Ni) is twice the total number of 12-inch

units (N2) ; hence,

/ ÍV2W \

^^==l(2Ñ2pi?j^^

_ niS2^

If we substitute this value of fh in the relative cost formula given

above

C2 n2d2 / niS2^\.

s^d2

In this example, a special study showed Si^ and «2^ to be 134.1 and

416.0 respectively, and the average times for locating the frame

and making the count for each size of frame were found to be
72 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

di — 94.36 and d2 = 129.00. Substituting these values in the

equation for relative cost,

Ci^ 4(134.1) (94.36)

C2 ~ (416.0) (129.00)

= 0.943

This result indicates that the 6-inch frame is slightly more eflScient

than the 12-inch frame.

In more general terms the cost of method 1 relative to the cost

of method 2 for a specified sampling error would be

The same result is obtained by thinking in terms of the relative

efficiency of the alternative procedures. As a measure of efficiency,

statisticians commonly use the reciprocal of the product of the cost

per unit and the squared coefficient of variation for the given

sample unit. If the coefficient of variation is symbolized by C and

the cost by d, the efficiency (U) is given by

U = -

The relative efficiency of two alternatives would then be

U2_ (di){C^)\ U^ _(¿2)(C2)"

Ui (d2)(C2)2^'^' U2 "(di)(Ci)2

In the previous example we had

(k = 94.36 Si2 ^ 134 1

dz = 129.00 §2^ = 416.0

For the 6-inch frame the squared coefficient of variation is

For the 12-inch frame the squared coefficient of variation would

be

The mean per unit for the 12-inch frame (^2) should be twice the

mean per unit for the 6-inch frame, so that we can write

^^'^ -(2iJ^-4^

Then the efficiency of the 12-inch frame relative to that of the 6-

inch frame is

U2 ^ 94.36 (134.1M^) _ 4(94.36) (134.1)

Ui 129.00 (416.0/4^i2) (129.00) (416.0)

= 0.943
ELEMENTARY FOREST SAMPLING 73

As before, the 6-inch frame appears more efficient than the 12-

mch frame.

Estimating Changes

Changes that have taken place in the characteristics of a forest

population are often of as much interest as their present status.

Periodic change in stand volume is, for example, a major concern

of foresters.

Estimating such changes usually requires sampling at the beginning and end of the period. The difference or some function of

the difference between the two estimates is the estimated change.

Ordinarily the same sampling method will be used each time, but

that is not absolutely necessary.

Temporary or permanent plots.—Estimating change by sampling at two different times always raises the question of temporary

or permanent sample plots. That is, should an entirely new set of

units be randomly selected for observation at each time, or should

the same units be observed at both times? A third alternative is

to have some temporary and some permanent plots in a double

sampling system : a large sample of temporary plots with a subsample of permanent plots.

The choice between temporary and permanent plots depends

heavily on the degree of correlation that can be expected between

the initial and final plot values. If a high positive correlation is

expected, permanent plots should give the better precision. If the

correlation is likely to be low or negative, temporary plots might

be better. If the period is relatively short and if cutting or heavy

mortality is unlikely, the correlation probably will be large and

positive, favoring the use of permanent plots. Where large volume

changes are likely to occur because of cutting, heavy mortality,

or a very long time interval, the correlation will be small or even

negative, favoring the use of temporary plots.

If there is enough information on cost and variability, the advantage of permanent plots with simple random sampling can be

weighed by computing the relative cost (Re) of obtaining a given

precision by the two methods.

Ko =

Cp(Si2+S22-25i2)

where: Ct = Cost of locating and making a single measurement

on a temporary plot.

Cp = Total cost of locating, measuring, monumenting, relocating, and remeasuring a permanent plot.

Si^ z= Variance among individual plots at the time of the

first measurement.

S2^ = Variance among individual plots at the time of the

second measurement.

Si2 = Covariance between the first and second measurements on individual plots.
74 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

If Ra is greater than 1, permanent plots should be used. If R^ is

less than 1, temporary plots will probably be better. Where remeasurements will be made several times, the average cost per

permanent plot will be reduced, swinging the ratio more favorably towards permanent plots.

Plot monumentation.—The question of kind and degree of plot

monumentation has been hotly debated among the users of permanent plots. Where any form of stand treatment is likely to

take place between measurements, it is generally conceded that the

plot location and form of monumentation should not be discernible

to those who make the stand treatments. It is very difficult, if not

humanly impossible, to avoid treating plot areas differently from

nonplot areas. At the same time, if the monuments are too cleverly

concealed, relocation costs will be increased and some plots may

not be found at all. Because the difficulty of plot relocation is

likely to be related to stand conditions that are in turn related

to growth, failure to relocate plots could slightly bias the estimates.

Sampling errors.—If the mean per unit at the time of the first

measurement is ^i, and the mean per unit at the time of the second

measurement is ^2» the estimated periodic change per unit is (^2

-Vi)'

With temporary plots, the standard error of the estimated

change would be

where Sy^^ and Sy^^ are the squared standard errors of the mean

at the time of the first and second measurements. The method of

computing s^j2 ^nd Sy^^ would be that appropriate to the particular sampling method used.

With permanent plots, the easiest procedure for computing the

standard error is to work with the individual differences. Thus, if

Vu stands for the first measurement of the i"» permanent plot and

t/2< stands for the second measurement on that plot, then di =

(2/2* -- Vu) ' The standard error of the mean difference is computed

from the di values with the formula appropriate for the particular sampling method.

Examples.—The above computations will be illustrated for a

simple random sample.

Temporary Plots

Initial observations: n = S

Vu = 12, 24, 27,14,16,10, 21,30

aiVu = 154 Vi = 19.25

s,,2 = 53.9286 5,,2 ^ V ^ 6.74
ELEMENTARY FOREST SAMPLING 75

Final observations: n = S

Vm = 27,18, 22,33,14, 26,16, 24

8

S 2/2/ = 180 ^2 = 22.50

3 =1

V ^ 40.0000 s^2^ = ^ = ^'^^

Then the estimated mean difference is

(^2 - ^i) = (22.50 - 19.25) = 3.25

The standard error of the mean difference is

s^y^-y^^ = V6.74 + 5.00

= 3.43

Permanent Plots

Permanent Plot No.

1X3^5678 Sum Mean

Initial observations {yu)- 24 14 16 27 10 30 12 21 154 19.25

Final observations {y^ù 26 18 22 27 14 33 16 24 180 22.50

Differences (i^i = 2/2i — 2/ii) 24604343 26 3.25

The estimated mean difference is

(^2 — yi) =U = 3.25

The standard error of the mean difference is calculated from the

di values with the formula for a simple random sample.

(»-1)

(22 4- 42 + . . . + 32) _ ^

= 3.0714

a.= ^Ï = 0.61.62

Design of Sample Surveys

It has been the purpose of this handbook to treat only one segment of the design of sample surveys, that of the sampling method
76 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

and associated computational procedures. These are the aspects

of sampling that seem to be most troublesome to foresters. But

several other phases of survey design also deserve attention. Some

of the points that should be considered in planning a survey are

summarized here.

The objective must he stated.—Specifically, identify the parameter to be estimated and the precision desired. An example of a

lucid objective might be: "To estimate the number of plantable

slash pine seedlings at the Riedsville Nursery. The estimate should

be within 1 percent of the true number, with 95-percent confidence." Vague statements ("To study the results of spraying..."

"To estimate the effectiveness of...") can and do result in an

appalling waste of survey efforts.

The population should he defined.—^What are the units constituting the population? What are the unit values? What units are

excluded from the population? Careful, accurate answers to these

questions will forestall numerous difficulties at later stages. A

generality worth repeating is that sampling design will be simplified if the specifications for the units used to define the population

are identical with those used in the sample. Even at that, the

definition and specification may be difficult. It may be easy to define a tree or a plot, but if a survey is to be made of farmers,

pulpwood contractors, or seed orchards, the unit may be very hard

to define. An attempt should be made to foresee the difficulties

that might arise in classifying a unit as in or out of the population ; the borderline instances will be a constant source of trouble

to enumerators and analysts.

The data to he collected should he specified.—Special attention

must be paid to getting all the data necessary to the objective.

It is a moot question how far one should go in taking supplementary data that is not pertinent to the main objective. Frequently

cooperators and reviewers, sensing an opportunity to obtain information on some pet project, will request that additional observations be made "while you're there." Such requests must be

carefully reviewed. "Free" information is not cheap if it is never

used or has an adverse effect on the main objective of the survey.

Measurement techniques must he prescribed.—The measurement

procedures should be stated unambiguously. The detail needed will

vary with the complexity of the measurements and the experience

of the personnel, but in general it is better to be annoyingly specific than trustingly vague. Terms such as merchantable top, overstory, undesirable, stocked, board-foot volume, and plantable

should be precisely defined.

The need for training and preliminary practice should be considered. And proficiency tests are not unwarranted—even for the

old hands who may have forgotten some of their earlier training

or developed bad habits.

The sampling units must be defined.—^Again, the totality of

sampling units, however distributed, must comprise the population. If the unit is obvious, e.g., a sawmill, no particular trouble

need arise. But if a variety of units are possible, a search of litera-
ELEMENTARY FOREST SAMPLING 77

ture will frequently uncover some profitable experience; if not,

re uirid optimum size and shape of sampling unit may be

The sampling method must be descnbed.—This handbook outlines a number of methods that have been found useful in forestry

1 nought, experience, and a review of literature will help in deciding which method is most appropriate for a particular situation.

Ihe method of selecting the sample units should be carefully

stated, and so should the procedure of locating the units in the

held. Saying that a two-stage design will be used with primaries

and secondaries selected at random is not enough. How will randomization be accomplished? And how will the unit be located in

the field? The possibilities of and antidotes for bias in locating

units deserve some thought. Timber cruisers will, for example,

tend to veer away from dense brush and openings when locating

plots by hand compass and pacing. House-to-house interviewers

have been known to neglect top-floor apartments and homes with

barking dogs.

At this stage it is also well to think out the procedures to be

used for estimating the parameters and sampling errors. Collecting data and then asking someone how to use it is a good way to

lose friends and waste survey money.

The sample size must be prescribed,—Once the desired precision, choice of sampling unit, and method of sampling have been

stated it is time to think of the size of sample. The sample should

be just large enough to give the specified precision, and no larger.

If the requisite information on costs and variances is available,

this decision should be made prior to the start of field work. In

the absence of such information, a preliminary survey may be

necessary.

Possible problems of data should be considered.—If the preceding steps are meticulously followed, problems arising at the datacollection stage are usually those of organization and personnel.

The greatest single stumbling block is the common failure of

supervisors to continue training and checking field crews or to

provide for editing of field forms. Some organizations find it

worthwhile to make punched-card sorts to check for recording

mistakes such as trees that are 3 inches in d.b.h. and have 14 logs

(instead of a 14-inch tree with 3 logs).

Data processing should be planned.—In most cases, procedures

for computation and analysis are fixed by the choice of sampling

methods. In organizing the computing, there may be some extraordinary considerations that merit early attention. If the volume

of data is small, computing may be readily absorbed in the daily

routine. If the volume is large, special staffing and special equipment may be desirable. If, for example, the analysis is to be on

electronic computers, it would be advisable to become familiar

with the special requirements necessary to electronic computing,

such as data format for keypunching, availability of programs,

and cost of programming.
78 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

REFERENCES FOR ADDITIONAL READING

Cochran, W. G.

1953. Sampling techniques. 330 pp., illus. Wiley, New York.

Deming, W. E.

1950. Some theory of sampling. 602 pp., illus. Wiley, New York.

Dixon, W. J., and Massey, F. J., Jr.

1957. Introduction to statistical analysis. Ed. 2, 488 pp., illus. McGrawHill, New York.

Hansen, M. H., Hurwitz, W. N., and Madow, W. G.

1953. Sample survey methods and theory. Vol. I, 638 pp., illus. Wiley,

New York.

Hendricks, W. A.

1956. The mathematical theory of sampling. 364 pp., illus. Scarecrow

Press, New Brunswick, N. J.

Schumacher, F. X., and Chapman, R. A.

1942. Sampling methods in forestry and range management. Duke Univ.

School Forestry Bui. 7, 213 pp., illus.

Snedecor, G. W1956. Statistical methods. Ed. 5, 534 pp., illus. Iowa State Univ. Press,

Ames, la.

Sukhatme, P. V.

1954. Sampling theory of surveys, with applications. 491 pp., illus. Iowa

State Univ. Press, Ames, la.

Yates, Frank.

1960. Sampling methods for censuses and surveys. Ed. 2, 440 pp., illus.

Hafner, New York.
PRACTICE PROBLEMS IN SUBSCRIPT AND

SUMMATION NOTATION

Values of the Variable Xij

j Classification (i==l,.. .,10) i Class1 2 3 4 5 6 7 8 9 10 ification

subtotals

1 6 4 2 0 4 3 5 9 6 8 47

1 p 2 4 8 4 2 111 6 2 1 30

1 :

3 2 3 2 8 4 8 2 1 1 2 33

*03 ?

4 1 0 3 2 0 0 0 2 4 8 20

g ^ 5 0 2 6 7 18 3 5 4 4 40

S 11 6 3 7 5 3 5 2 4 3 2 6 40

7 2 17 2 6 11 6 4 3 33

j Classifica

subtotal

tion

s

18 25 29 24 21 23 16 32 23 32 243

Examples:

a:i.3 = 2 íTT.S = 6 0^3.7 = 2 íTé.T = 0

7 10

EU'" = (íCi.i + a^i.a + . . . + ^1.10 + í»2.1 + 3^2.2

+ . . . + aÍT.lo)

= (6 +4+... +8+ 4+ 8+ ...+3)

= 243

3 3

= (a;2.1 + í^,2 + «^2,3 + ÍC3,1 + 3^3,2 + ^3,3)

^ (4 _|_ 8.+4 + 2 + 3 + 2) = 23

2 4

i=l 3=3

= (Xl,s' + X,,^ + «^2.3^ + X2.é^)

i=2 V=3 /

= (22 4- 02 + 42 + 22) =: 24

= (a;2.3 + X2A)^+ (0^3.3 + 0:3.4)'

= (4 + 2)2+ (2 + 8)2 = 136

79
80 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

(6 9 \ 2

S S Xij ) = (íCs.S + ÍC5,9 + XQ,8 + XQ,íí)^

i=5 3=8 /

= (5 4-4 + 3 + 2)2r=:196

s X3J = (X3,i + a;3,2 + ... + XB,IO)

= (2 4- 3 + ... + 2) == 33

S aîie^ = (32 + 12 + 82 4-... + 12) = 143

(s..)^

», Í

= 292 ^ g4i

S Xii = 243

7

S Xi2Xis = (a;i.2) (aîi.s) + (0:2,2) (0:2.3)

'-^ + •.. + (2:7.2) (0:7.3)

=: (4) (2) + (8) (4) + ... + (1) (7) = 100

S(0:5/ — x^i) = (0:5.1 — 0:4.1) + (0:5,2 — 0:4,2)

' + ... + (0:5.10 — 0:4.10)

= ÍS0:5;—S0:4/j

= (40 - 20) = 20

S(o:5i-o:4;)2 = (0-1)2 4.(2-0)2+ (6-3)2

+ ...+ (4-8)2

= 138

S 0:5/ -S 0:4/ == (02 4- 22 4- ... 4- 42) - (12 4. 02

^ i +... + 82) = 122

(s0:5/)' -(s 0:4/)' = (402 - 202)

= 1,200

S (0:5/ — 0:4;) = S0:5/ — S0:4/

= [40-20]2 = 400
J2BX2,

ELEMENTARY FOREST SAMPLING 81

= 3(a;2.i) +3(0:2.2) + ... + 3(0:2,10)

= 3(o;2,i + o;2,2 + ...+ 0^2,10)

= 3 (^X2Í\=S(S0) =:90

S (x^j - 6) = (0:4.1 - 6) + (0:4 2 - 6)

+ ... + (0:4.10 - 6)

= (Xé,l + 0:4,2 + ... + 0:4.10)

-6-6-...-6

= íso:4i) -10(6)

= (20 - 60) = -40
TABLES

TABLE 1.—Ten thov^and randomly assorted digits

54463

15389

85941

61149

05219

41417

28857

17783

40960

96754

84357

06818

62111

47534

98614

24856

96887

90801

65166

75884

16777

46230

42902

81007

85205

40756

69440

81619

94070

00015

87403

62820

09243

75993

08648

12479

21472

77812

12952

87116

48877

10-14 15-19

66905

18850

82414

11286

10651

87719

20652

10806

29881

66164

68864

49927

07248

67879

84460

44898

80621

42816

84S18

58660

85966

41180

44105

71726

57715

79931

00544

09351

66223

77408

95108

42968

88877

01432

20-24 25-29 30-84

13858

58925

92511

46614

16249

91630

62800

10089

47861

45690

50423

89292

23410

59844

98795

86085

37390

28420

72305

21460

89380

94710

10154

67382

90669

78030

50948

75019

36466

67372

84767

12740

14922

18644

78286

76766

70219

23474

95425

29085

96325

16269

52862

84502

64886

21145

39981

84740

78258

86679

60332

63116

85693

02540

48730

39766

02432

62616

91318

01175

91380

20423

23248

65978

62733

72095

20002

05217

62481

73947

54440

78448

71068

63342

32141

41943

89872

87894

03164

60137

19774

01385

33451

83463

97365

47286

49177

77379

53249

90600

21605

48167

90368

42846

80268

47366

45375

81378

08186

26927

15345

77455

75577

30976

76305

75779

90279

37231

27083

71113

80182

11661

13491

84770

44104

94771

18106

41067

85436

10620

69337

18119

49087

59391

99567

10363

86859

11258

96068

54463

16874

92494

15669

99116

15696

97720

11666

71628

40501

22618

75112

57430

73528

25991

55-59 60-64

76364

97518

19558

24591

47237

62677

63157

56689

75486

10703

16369

13841

73130

51089

65576

30485

02671

45648

52098

77204

51400

64432

35911

73800

57412

76593

35682

65178

51269

71681

78783

98215

62173

98191

02146

10421

34434

70769

09134

96667

82718

04615

25670

16706

55368

14530

91017

13215

91316

40844

23476

90637

02132

84342

05597

05540

88596

64721

70-74 75-79

87024

27062

98342

99612

31721

31389

03505

53256

52967

63110

03388

35979

41632

41995

10798

14878

90813

43648

64086

86413

82848

96621

61891

59798

94335

80428

71824

62233

72389

81872

67104

17622

13699

39719

09847

49268

81366

75888

71693

33475

48472

>

84 85-89 90-94 95-99

04190

43918

27101

32803

39936

83671

80827

96363

35213

39495

53988

33423

73631

36684

22281

95441

34598

66049

43132

42740

39318

24696

96574

01896

37855

67708

02566

31855

39892

73917

52887

0S840

39100

71087

67453

07449

18707

69361

67466

16783

15496

72856

21511

14414

06175

35434

77515

90464

83991

06235

15297

80972

34334

60518

82802

01087

34471

17217

84148

43269

47985

85489

06375

20168

66762

47676

79949

82758

24057

09577

This table is reproduced, by permission of the author and publishers, from table 1.5.1 of Snedecor's StcUistical Methods (5th ed.), Iowa State University Press.
TABLE 1.—Ten thousand randomly assorted digits (continued)

00-04 05-09 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-^9 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94

20411

58212

70577

94522

42626

16051

08244

59497

97155

98409

45476

89300

50051

31753

79152

44560

83544

91621

91896

55751

85156

07521

01122

67081

13160

27647

04392

13428

66162

84882

69700

95137

85178

38750

83378

38689

86141

00881

67126

62515

87689

51111

89950

06468

24969

71659

85651

57194

33851

09419

40293

95763

65109

50741

91631

31310

77250

83635

15707

04900

04151

21108

95493

12236

72373

16944

15718

61210

62038

88678

16752

44705

89964

09985

47420

96597

30329

66315

89642

20190

56540

71381

54224

03795

82627

76046

79643

17401

54450

94211

51211

58434

20792

25930

11658

91428

56535

64900

39564

30459

46177

59077

02263

00664

39102

74373

87687

76999

67699

79169

03252

19031

46716

04894

01412

61527

66790

23166

12275

02306

18760

42912

05615

55017

62315

42054

44741

99547

58580

11738

72882

69124

20441

65706

05400

24816

24617

69942

13953

42451

20781

08467

17852

12630

37204

55539

12239

97017

87236

68680

12696

05437

32404

47629

55784

17805

82171

39435

77448

79149

64559

27491

98375

17771

07105

41273

77054

96739

93758

39038

17918

54132

95374

21896

59058

11859

53634

48708

71710

33278

18710

97501

26333

93842

89415

52068

30506

69448

11844

60631

72655

41567

22557

03887

33258

22716

48805

68618

65747

91777

55349

23466

60142

09808

87530

01117

83266

76970

37074

83712

20287

74261

64081

05617

26793

27366

56760

77888

28440

16738

59348

12900

75086

99495

26075

13636

65198

06514

56862

32592

49863

75818

74951

72850

42271

10909

43338

38100

07819

94441

93410

45030

60159

11695

71775

23537

61484

31671

44785

30101

69727

86538

08478

47760

95466

48737

44300

98147

21580

77033

16369

96279

07425

45761

29845

49939

29181

45386

23377

15679

39516

68624

78295

94443

27041

96001

67814

74307

54719

73399

34736

58904

58103

51459

12147

89033

14709

62369

15865

60774

38155

79152

54656

64936

69543

47961

47971

52372

07515

74739

94924

13484

38190

93459

95126

29793

74798

84481

85417

86532

14810

10526

42664

01596

95256

23943

83841

49955

47231

87832

82721

05672

21810

97588

42563

48599

61496

97610

43189

27227

07571

70545

66192

85515

03845

73467

12731

11231

25878

02736

37875

66990

09054

78735

60048

06158

89755

44464

23746

29226

31776

50803

71153

20271

38636 33717 67698

52022

42474

16255

73579

46703

72781

39286

59064

27058

05497

03134

05192

50771

65938

55903

23608

96071

05383

72744

21315

65128

52125

41330

45141

91077

60651

17777

92359

98265

72606

59566

65340

07210

40467

33625

70322

48657

83665

81581

44115

15873

05813

39902

88208

00132

14551

82521

35234

40197

91321

42338

This table is reproduced, by permission of the author and publishers, from table 1.5,1 of Snedecor's Statistical Methods {5th ed.), Iowa State University Press.
TABLE 1.—Ten thoiisand randomly assorted digits (continued)

00-04 05-09 10-14 15-19 20-24 25-29 30-34 35-39 40-44 46-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99 g

05845

74897

20872

48194

11303

54374

42587

40177

79788

40538

64016

49767

76974

68978

S6444

63664

44249

00512

68373

54570

96156

61605

57790

87118

67826

34421

20384

37065

98690

76128

68248

79000

73598

12691

66108

08480

70551

98600

87800 07391

26188 86647

39662

04050

78630

67369

36017

89177

01240

79970

81471

16947

61046

09491

24526

97161

18609

17903

29795

40646

48174

66328

51014

88132

75641

00660

33106

90849

91588

72602

73150

93871

08404

96026

78088

14971

11694

97306

65570

18116

33510

25730

81355

05873

86904

08665

78371

97720

57689

84633

24141

27084

42274

62463

99721

82684

60177

98573

25325

21196

04558

31741

44072

13568

48119

98131

67588

04838

14748

23489

33102

79109

00497

64610

79848

00427

00781

61463

07294

40192

91705

17056

86723

77243

76082

52503

49416

97191

30376

37292

62036

40254

17620

34602

46206

09426

61126

99425

31778

62073

32660

67842

84149

51153

72506

91691

79172

24130

44448

63798

21753

76970

05967

49967

96401

76608

87440

26904

79936

29656

64280

67158

46797

11397

42607

67913

96767

07419

67450

61446

18847

82487

58212

67071

64600

77212

93448

21627

27029

27928

10608

36894

61946

67042

76013

66671

21641

23037

24768

73041

24210

32847

16916

66176

46299

22847

41851

28444

47520

34978

37404

82400

90814

19192

77685

23757

46989

92970

74346

87646

50099

10127

67995

26304

31282

00041

84047

13385

47839

54160

59497

62378

63271

80416

09127

14833

82756

52593

16364

96257

94243

59596

41309

71038

46900

81977

80217

21005

12180

45386

91586

98856

13142

52099

51718

98021

08759

20563

56612

05096

23860

07316

40088

27636

45146

64984

18984

84934

27137

16861

95917

83174

70663

03871

74645

68446

95766

03192

26216

41467

98176

45153

06146

76348

64091

82667

69214

14253

03191

38043

47526

34803

68553

13088

05271

49486

74648

11623

27789

05046

65376

10019

62386

23309

64837

17896

65211

04115

02785

70381

76582

48970

69292

54098

92479

28639

16561

58444

94066

88914

21526

52406

94770

99429

33624

27762

35397

78286

12092

64625

62675

45683

06466

68559

06490

76610

66095

79820

44832

73064

85332

07426

07255

43169

68774

55849

71160

34174

26679

44984

74976

69597

73002

86606

65091

26119

18877

50264

61553

77753

66259

60013

97144

36104

91018

92426

64777

11130

49307

70066

63623

55710

19456

31220

19099

06340

97786

35515

80407

08187

16742

37655

79085

78195

83378

91994

51254

62717

15478

71551

03591

30180

73040

43816

58136

96459

29315

14032

48885

99761

59180

62556

64524

48109

This table is reproduced, by permission of the author and publishers, from table 1.5.1 of Snedecor's Statistical Methods (5th ed.), Iowa State University Press.
TABLE 1.—Ten thousand randomly assorted digits (continued)

00-04 05-09 10-14 15-19 20-24 25-29 30-34 135-39 40-44 45^9 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

64437

91714

20902

12217

45177

28325

29019

84979

50371

53422

67453

07294

79544

64144

95 90919

06670

36634

75101

05112

55809

19843

51564

99566

S1890

65162

32242

53662

17646

86007

02863

90814

28776

81353

06825

35651

86353

00302

85442

11883

57353

93976

72891

71222

47982

69122

38040

14742

95712

32245

48431

28373

31391

70371

42307

08804

66116

56219

48513

69711

89816

74819

45338

58318

86275

52062

85745

72654

41968

42502

39418

05028

08279

82279

04835

34333

81459

52281

63571

52746

64791

67062

67950

41620

23446

16015

46471

00042

92276

83678

67106

51583

49915

30033

91794

79256

39070

55791

33816

14510

47913

64604

26146

11158

64716

32048

68237

66613

24162

62402

77591

41256

26010

05228

19000

94889

94068

69702

74768

03444

76094

74921

54577

08815

82567

25563

70225

07202

88968

39500

28210

46924

60948

62107

62056

58050

53381

49337

81508

51144

55743

96679

17735

47525

46049

49581

47597

99515

14695

87351

34075

60839

18685

60885

57390

91486

72998

28666

88674

18827

74701

54863

42201

77705

71186

14124

18431

46378

33137

33272

56437

48992

37503

42746

19180

99942

79952

75787

35355

56705

22390

10704

58861

78339

80540

95330

34650

46240

92978

99878

31443

53809

77716

42833

00840

03183

19462

55461

39272

15100

10515

67849

06118

62246

54721

21866

14994

92973

11591

61130

51445

26685

79696

71875

73268

13191

96062

71213

81994

59537

51228

18682

63571

68927

56401

24333

17025

02804

31121

97867

63873

35531

41070

34662

10937

87995

97938

81142

56134

74530

81038

32679

56492

84202

08253

03879

01785

06682

47266

56641

86746

26263

55571

19162

56642

79631

09474

67582

38004

07661

68416

00608

86406

25695

64091

89403

81460

07828

75174

89712

92557

90102

90915

25371

72942

62272

19850

96062

91837

02051

17577

14621

31229

66212

47831

42272

79460

68153

77642

31356

46287

73090

03785

74021

67599

80161

49430

75999

91332

77511

80721

09975

91403

54016

55436

33452

90257

16175

50865

89094

24471

87320

22311

44540

63956

24311

16197

13513

06118

95007

21950

67206

42212

05134

05500

91588

98475

91853

97036

08452

97577

74087

57257

78742

45148

86197

06047

86192

87644

06140

80744

76884

08424

32341

62862

99304

22343

82244

64896

64158

72492

10918

69008

16846

99046

21296

42694

27612

62700

07716

37635

81450

41587

55111

79614

71402

49872

79846

47493

77656

97528

S0144

16162

64809

43325

12648

73757

71775

99121

30030

78235

76287

41915

44103

54809

99681

05941

45447

This table is reproduced, by permission of the author and publishers, from table 1.5.1 of Snedecor's Statiètical Methods (5th ed.), Iowa State University Press.
86 AGRICULTURE HANDBOOK 232, U.S. DEPT. OP AGRICULTURE

TABLE 2.—The distribution of t

df

Probability

.5 .4 .3 .2 .1 .05 .02 ,01 .001

1.-.. 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.619

2..-. .816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.598

3.... .765 .978 1.250 1.638 2.353 3.182 4.541 5.841 12.941

4.... .741 .941 1.190 1.533 2.132 2.776 3.747 4.604 8.610

5.... .727 .920 1.156 1.476 2.015 2.571 3.365 4.032 6.859

6.... .718 .906 1.134 1.440 1.943 2.447 3.143 3.707 5.959 7.— .711 .896 1.119 1.415 1.895 2.365 2.998 3.499 5.405 8...- .706 .889 1.108 1.397 1.860 2.306 2.896 3.355 5.041 9..-. .703 .883 1.100 1.383 1.833 2.262 2.821 3.250 4.781 10.... .700 .879 1.093 1.372 1.812 2.228 2.764 3.169 4.587

11.... .697 .876 1.088 1.363 1.796 2.201 2.718 3.106 4.437 12.... .695 .873 1.083 1.356 1.782 2.179 2.681 3.055 4.318 13.... .694 .870 1.079 1.350 1.771 2.160 2.650 3.012 4.221 14.... .692 .868 1.076 1.345 1.761 2.145 2.624 2.977 4.140 15.... .691 .866 1.074 1.341 1.753 2.131 2.602 2.947 4.073

16.... .690 .865 1.071 1.337 1.746 2.120 2.583 2.921 4.015 17.... .689 .863 1.069 1.333 1.740 2.110 2.567 2.898 3.965 18.-.. .688 .862 1.067 1.330 1.734 2.101 2.552 2.878 3.922 19.... .688 .861 1.066 1.328 1.729 2.093 2.539 2.861 3.883 20.... .687 .860 1.064 1.325 1.725 2.086 2.528 2.845 3.850

21.... .686 .859 1.063 1.323 1.721 2.080 2.518 2.831 3.819 22.-.. .686 .858 1.061 1.321 1.717 2.074 2.508 2.819 3.792 23.... .685 .858 1.060 1.319 1.714 2.069 2.500 2.807 3.767 24.... .685 .857 1.059 1.318 1.711 2.064 2.492 2.797 3.745 25-.-. .684 .856 1.058 1.316 1.708 2.060 2.485 2.787 3.725

26.... .684 .856 1.058 1.315 1.706 2.056 2.479 2.779 3.707

27.... .684 .855 1.057 1.314 1.703 2.052 2.473 2.771 3.690

28.... .683 .855 1.056 1.313 1.701 2.048 2.467 2.763 3.674

29.... .683 .854 1.055 1.311 1.699 2.045 2.462 2.756 3.659

30.... .683 .854 1.055 1.310 1.697 2.042 2.457 2.750 3.646

40-... .681 .851 1.050 1.303 1.684 2.021 2.423 2.704 3.551

60.... .679 .848 1.046 1.296 1.671 2.000 2.390 2.660 3.460

120.... .677 .845 1.041 1.289 1.658 1.980 2.358 2.617 3.373

00 -. .674 .842 1.036 1.282 1.645 1.960 2.326 2.576 3.291

This table is abridged from table III of Fisher and Yates' Statistical Tables

for Biological, Agricultural, and Medical Research, Oliver and Boyd Ltd.,

Edinburgh. Permission has been given by the authors and publishers.
ELEMENTARY FOREST SAMPLING 87

TABLE 3.—Confidence intervals for binominal distribution

95-percent interval

Number

observed

(/)

Size of sample, n

Fraction

observed

fin

Size of sample

10 15 20 80 60 100 250 1000

0 0 31

0 45

3 56

7 65

12 74

19 81

26 88

35 93

44 97

55 100

69 100

0

0

2

4

8

12

16

21

27

32

38

45

52

60

68

78

22

32

40

48

65

62

68

73

79

84

88

92

.96

98

100

100

0

0

1

8

6

9

12

15

19

23

27

32

36

41

46

51

56

62

69

75

83

17

25

81

38

44

49

64

69

64

68

73

77

81

85

88

91

94

97

99

100

100

0

0

1

2

4

6

8

10

12

15

17

20

23

25

28

31

34

37

40

44

47

50

54

57

61

65

69

73

78

83

88

12

17

22

27

81

35

39

43

46

60

63

66

60

63

66

69

72

75

77

80

88

85

88

90

92

94

96

98

99

100

100

0

18

20

21

23

25

27

28

30

32

34

36

37

39

41

43

45

47

50

52

54

56

57

59

62

64

66

69

71

73

76

78

81

83

86

89

93

07

11

14

17

19

22

24

27

29

31

34

86

38

41

43

44

46

48

60

63

56

67

59

61

63

64

66

68

70

72

73

75

77

79

80

82

84

85

87

88

90

91

98

94

95

97

98

99

100

100

100

0

0

0

1

1

2

2

8

4

4

5

6

6

7

8

9

9

10

11

12

13

14

14

15

16

17

18

19

19

20

21

22

23

24

25

26

27

28

28

29

30

31

32

33

34

35

36

37

38

39

40

4

5

7

8

10

11

12

14

16

16

18

19

20

21

22

24

25

26

27

28

29

80

31

32

83

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

60

61

52

63

54

65

56

57

58

59

60

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

.20

.21

:S

.24

.25

.26

.27

.28

.29

.30

.31

.32

.88

.34

.35

.86

.37

.38

.39

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

.50

0

0

1

1

2

3

3

t

6

7

7

8

9

10

10

11

12

13

14

16

16

17

18

19

20

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

1

4

5

6

7

9

10

11

12

13

14

16

17

18

19

20

21

22

23

24

26

27

28

29

30

31

82

33

84

86

36

37

38

39

40

41

42

43

44

46

46

47

48

49

60

51

62

63

54

55

56

0

0

1

2

3

4

5

6

6

7

8

9

10

11

12

13

14

15

16

17

18

19

19

20

21

22

23

24

26

26

27

28

29

30

31

32

33

34

35

36

87

38

39

40

41

42

43

44

45

46

47

0

1 2

2 3

3 4

4 5

5 7

6 8

7 9

8 10

9 11

10- 12

11.._ 13

12 14

13 15

14 16

15 17

16 18

17 19

18 21

19 22

20 23

21 24

22 - -..

25

23 26

24 27

25 28

26 29

27 30

28 31

29 32

30 33

31 34

82 35

83 36

34 37

35 38

36 39

37 40

38 41

39 42

40 43

41 44

42 45

43 46

44 47

45

48

46

49

47

50

48

51

49

62

50

53

1

This table is reoroduced, by permission of the author and publishers, from table 1.3.1 of This table ^|^^JP¿°°;Í3^«^¿,¿¿jg„j Methods (ed. 5), Iowa-State Umversity Press.
88 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

TABLE 3.—Confidence intervals for binominal distribution

(continued)

Number

observed (/)

)-percent interval

s of sample, n

13

19

26

35

46 100

59 100

99

100

100

99 S

100 3

100 3

100 3

40

41

42

43

44

45

46

47

48

49

50

51

62

53

54

55

66

57

58

59

60

61

Fraction

observed

fin

0.00

.01

.02

Size of sample

0

0

1

1

.04

.06 2

3

3

4

6

.10

.11

.12

.13

.14

.16

.16

.17

.18

.19

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

.60

3

4

6

6

7

8

9

9

10

11

12

13

14 20

16 21

16 22

17 23

18 24

19 26

2 21

3 22

31

24 32

65 4

56 4

57 4

58 46

}44

7 46
ELEMENTARY FOREST SAMPLING 89

TABLE 4.—Arcsin transformation (angles corresponding to percentages, angle = arcsin ^/percentage)

% 0 1 2 3 4 5 6 7 8 9

0.0 0 0.57 0.81 0.99 1.15 1.28 1.40 1.52 1.62 1.72

0.1 1.81 1.90 1.99 2.07 2.14 2.22 2.29 2.36 2.43 2.50

0.2 2.56 2.63 2.69 2.75 2.81 2.87 2.92 2.98 3.03 3.09

0.3 3.14 3.19 3.24 3.29 3.34 3.39 3.44 3.49 3.53 3.58

0.4 3.63 3.67 3.72 3.76 3.80 3.85 3.89 3.93 3.97 4.01

0.5 4.05 4.09 4.13 4.17 4.21 4.25 4.29 4.33 4.37 4.40

0.6 4.44 4.48 4.52 4.55 4.59 4.62 4.66 4.69 4.73 4.76

0.7 4.80 4.83 4.87 4.90 4.93 4.97 5.00 5.03 5.07 5.10

0.8 5.13 5.16 5.20 5.23 5.26 5.29 5.32 5.35 5.38 5.41

0.9 5.44 5.47 5.50 5.53 5.56 5.59 5.62 5.65 5.68 5.71

1 ... 5.74 6.02 6.29 6.55 6.80 7.04 7.27 7.49 7.71 7.92

2 8.13

9.98

8.33

1,0.14

8.53

10.31

8.72

10.47

8.91

10.63

9.10

10.78

9.28

10.94

9.46

11.09

9.63

11.24

9.81

3 11.39

4 11.54 11.68 11.83 11.97 12.11 12.25 12.39 12.52 12.66 12.79

5 12.92 13.05 13.18 13.31 13.44 13.56 13.69 13.81 13.94 14.06

6 14.18

15.34

14.30

15.45

14.42

15.56

14.54

15.68

14.65

15.79

14.77

15.89

14.89

16.00

15.00

16.11

15.12

16.22

15.23

7 16.32

8 16.43 16.54 16.64 16.74 16.85 16.95 17.05 17.16 17.26 17.36

9 17.46 17.56 17.66 17.76 17.85 17.95 18.05 18.15 18.24 18.34

10 18.44 18.53 18.63 18.72 18.81 18.91 19.00 19.09 19.19 19.28

11 19.37 19.46 19.55 19.64 19.73 19.82 19.91 20.00 20.09 20.18

12.. 20.27 20.36 20.44 20.53 20.62 20.70 20.79 20.88 20.96 21.05

13 21.13 21.22 21.30 21.39 21.47 21.56 21.64 21.72 21.81 21.89

14 21.97 22.06 22.14 22.22 22.30 22.38 22.46 22.55 22.63 22.71

15 22.79 22.87 22.95 23.03 23.11 23.19 23.26 23.34 23.42 23.50

16 23.58 23.66 23.73 23.81 23.89 28.97 24.04 24.12 24.20 24.27

17 . 24.35 24.43 24.50 24.58 24.65 24.73 24.80 24.88 24.95 25.03

18 25.10 25.18 25.25 25.33 25.40 25.48 25.55 25.62 25.70 25.77

19 25.84 25.92 25.99 26.06 26.13 26.21 26.28 26.35 26.42 26.49

20 26.56 26.64 26.71 26.78 26.85 26.92 26.99 27.06 27.13 27.20

21 27.28 27.35 27.42 27.49 27.56 27.63 27.69 27.76 27.83 27.90

22 27.97 28.04 28.11 28.18 28.25 28.32 28.38 28.45 28.52 28.59

23 28.66 28.73 28.79 28.86 28.93 29.00 29.06 29.13 29.20 29.27

24 29.33 29.40 29.47 29.53 29.60 29.67 29.73 29.80 29.87 29.93

25 30.00 30.07 30.13 30.20 30.26 30.33 30.40 30.46 30.53 30.59

26 30.66 30.72 30.79 30.85 30.92 30.98 31.05 31.11 31.18 31.24

27 31.31 31.37 31.44 31.50 31.56 31.63 31.69 31.76 31.82 31.88

28 31.95 32.01 32.08 32.14 32.20 32.27 32.33 32.39 32.46 32.52

29 32.58 32.65 32.71 32.77 32.83 32.90 32.96 33.02 33.09 33.15

30 .. 33.21 33.27 33.34 33.40 33.46 33.52 33.58 33.65 33.71 33.77

31 33.83 33.89 33.96 34.02 34.08 34.14 34.20 34.27 34.33 34.39

32 34.45 34.51 34.57 34.63 34.70 34.76 34.82 34.88 34.94 35.00

33 35.06 35.12 35.18 35.24 35.30 35.37 35.43 35.49 35.55 35.61

34 35.67

36.27

35.73

36.33

35.79

36.39

35.85

36.45

35.91

36.51

35.97

36.57

36.03

36.63

36.09

36.69

36.15

36.75

36.21

35 36.81

36 36.87 36.93 36.99 37.05 37.11 37.17 37.23 37.29 37.35 37.41

37 37.47 37.52 37.58 37.64 37.70 37.76 37.82 37.88 37.94 38.00

38 38.06 38.12 38.17 38.23 38.29 38.35 38.41 38.47 38.53 38.59

39.. 38.65 38.70 38.76 38.82 38.88 38.94 39.00 39.06 39.11 39.17

40 39.23

39 82

39.29

39.87

39.35

39.93

39.41

39.99

39.47

40.05

39.52

40.11

39.58

40.16

39.64

40.22

39.70

40.28

39.76

40.34 41

42

43

44

40 40 40.46 40.51 40.57 40.63 40.69 40.74 40.80 40.86 40.92

40.98 41.03 41.09 41.15 41.21 41.27 41.32 41.38 41.44 41.50

41.55 41.61 41.67 41.73 41.78 41.84 41.90 41.96 42.02 42.07
90 AGRICULTURE HANDBOOK 232, U.S. DEPT. OF AGRICULTURE

TABLE 4.—Arcsin transformation (angles corresponding to percentages, angle = arcsin ^percentage) (continued)

45-.

46..

47..

48...

49...

50...

51...

52...

53...

54...

55...

56...

57...

58...

59...

60...

61...

62...

63...

64...

65...

67....

68....

69....

70....

71....

72....

73....

74....

75....

76....

77....

78....

79....

80....

81....

82....

83....,

84.....

85

86.....

87

88

89

90..

91..

92..

93..

94..

95..

96..

97..

0

42.13

42.71

43.28

43.85

44.43

45.00

45.57

46.15

46.72

47.29

47.87

48.45

49.02

49.60

50.18

50.77

51.35

51.94

52.53

53.13

53.73

54.33

54.94

55.55

56.17

56.79

57.42

58.05

58.69

59.34

60.00

60.67

61.34

62.03

62.72

63.44

64.16

64.90

65.65

66.42

67.21

68.03

68.87

69.73

70.63

71.56

72.54

73.57

74.66

75.82

77.08

78.46

80.02

81.87

42.19

42.76

43.34

43.91

44.48

45.06

45.63

46.20

46.78

47.35

47.93

48.50

49.08

49.66

50.24

50.83

51.41

52.00

52.59

53.19

53.79

54.39

55.00

55.61

56.23

56.85

57.48

58.12

58.76

59.41

60.07

60.73

61.41

62.10

62.80

63.51

64.23

64.97

65.73

66.50

67.29

68.11

68.95

69.82

70.72

71.66 I

72.64

73.68

74.77

75.94

42.25

42.82

43.39

43.97

44.54

45.11

45.69

46.26

46.83

47.41

47.98

48.56

49.14

49.72

50.30

50.89

51.47

52.06

52.65

53.25

53.85

54.45

55.06

55.67

56.29

56.91

57.54

58.18

58.82

59.47

60.13

60.80

61.48

62.17

62.87

63.58

64.30

65.05

65.80

66.58

67.37

68.19

69.04

69.91

70.81

71.76

72.74

73.78

74.88

76.06

42.30

42.88

43.45

44.03

44.60

45.17

45.75

46.32

46.89

47.47

48.04

48.62

49.20

49.78

50.36

50.94

51.53

52.12

52.71

53.31

53.91

54.51

55.12

55.73

56.35

56.98

57.61

58.24

58.89

59.54

60.20

60.87

61.55

62.24

62.94

63.65

64.38

65.12

65.88

66.66

67.45

68.28

69.12

70.001

70.91

71.85

72.84

73.89

75.00

76.19

77.21

78.61

77.34 77.48

78.76 78.91

42.36

42.94

43.51

44.08

44.66

45.23

45.80

46.38

46.95

47.52

48.10

48.68

49.26

49.84

50.42

51.00

51.59

52.18

52.77

53.37

53.97

54.57

55.18

55.80

56.42

57.04

57.67

58.31

58.95

59.60

60.27

60.94

61.62

62.31

63.01

63.72

64.45

65.20

65.96

66.74

67.54

68.36

69.21

70.09

71.00

42.42

42.99

43.57

44.14

44.71

71.95

72.95

74.00

75.11

76.31

77.61

79.06

45.29

45.86

46.43

47.01

47.58

48.16

48.73

49.31

49.89

50.48

51.06

51.65

52.24

52.83

53.43

54.03

54.63

55.24

55.86

56.48

57.10

57.73

58.37

59.02

59.67

60.33

61.00

61.68

62.37

63.08

63.79

64.52

65.27

66.03

66.81

67.62

68.44

69.30

70.18

71.09

72.05

73.05

74.11

75.23

76.44

6

42.48

43.05

43.62

44.20

44.77

45.34

45.92

46.49

47.06

47.64

48.22

48.79

49.37

49.95

50.53

51.12

51.71

52.30

52.89

53.49

54.09

54.70

55.30

55.92

56.54

57.17

57.80

58.44

59.08

59.74

60.40

61.07

61.75

62.44

63.15

63.87

64.60

65.35

66.11

66.89

67.70

68.53

69.38

70.27

71.19

72.15

73.15

74.21

75.35

76.56

42.53

43.11

43.68

44.25

44.83

45.40

45.97

46.55

47.12

47.70

48.27

48.85

49.43

50.01

50.59

51.18

51.77

52.36

52.95

53.55

54.15

54.76

55.37

55.98

56.60

57.23

57.86

58.50

59.15

59.80

60.47

61.14

61.82

62.51

63.22

63.94

64.67

65.42

66.19

66.97

67.78

68.61

69.47

70.36

71.28

8

42.59

43.17

43.74

44.31

44.89

45.46

46.03

46.61

47.18

47.75

48.33

48.91

49.49

50.07

50.65

51.24

51.83

52.42

53.01

53.61

54.21

54.82

55.43

56.04

56.66

57.29

57.92

58.56

59.21

59.87

60.53

61.21

61.89

62.58

63.29

64.01

64.75

65.50

66.27

67.05

9

72.24

73.26

74.32

75.46

76.69

67.86

68.70

69.56

70.45

71.37

72.34

73.36

74.44

42.65

43.22

43.80

44.37

44.94

45.52

46.09

46.66

47.24

47.81

48.39

48.97

49.54

50.13

50.71

51.30

51.88

52.48

53.07

53.67

54.27

54.88

55.49

56.11

56.73

57.35

57.99

58.63

59.28

59.93

60.60

61.27

61.96

62.65

63.36

64.08

64.82

65.57

66.34

67.13

67.94

68.78

69.64

70.54

71.47

72.44

73.46

74.55

77.75

79.22

80.19 80.37 80.54 80.72 80.90

77.89

79.37

81.09

78.03

79.53

81.28

82.08 82.29 82.51 82.73 82.96 83.20 83.45

75.58 75.70

76.82 76.95

78.17 78.32

79.69 79.86

81.47 81.67

83.71 83.98
ELEMENTARY FOREST SAMPLING 91

TABLE 4.—Arcsin transformation (angles corresponding to percentages, angle = arcsin ^/percentage) (continued)

% 0 1 2 3 4 5 6 7 8 9

99.0

99.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100.0

84.26

84.56

84.87

85.20

85.56

85.95

86.37

86.86

87.44

88.19

90.00

84.29

84.59

84.90

85.24

85.60

85.99

86.42

86.91

87.50

88.28

84.32

84.62

84.93

85.27

85.63

86.03

86.47

86.97

87.57

88.38

84.35

84.65

84.97

85.31

85.67

86.07

86.51

87.02

87.64

88.48

84.38

84.68

85.00

85.34

85.71

86.11

86.56

87.08

87.71

88.60

84.41

84.71

85.03

85.38

85.75

86.15

86.61

87.13

87.78

88.72

84.44

84.74

85.07

85.41

85.79

86.20

86.66

87.19

87.86

88.85

84.47

84.77

85.10

85.45

85.83

86.24

86.71

87.25

87.93

89.01

84.50

84.80

85.13

85.48

85.87

86.28

86.76

87.31

88.01

89.19

84.53

84.84

85.17

85.52

85.91

86.33

86.81

87.37

88.10

89.43

This table is reproduced, by permission of the author and publishers, from

table 11.12.1 of Snedecor's Statistical Methods (ed. 5), Iowa State University

Press. Permission has also been granted by the original author, Dr. C. I. Bliss,

of the Connecticut Agricultural Experiment Station.

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