A UNITED STATES DEPARTMENT OF COMMERCE

PUBLICATION

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National Bureau of Stan library, £.0I Admjn

APR 2 1973 NBS TECHNICAL NOTE 756

A Users' Guide to the OMNITAB

Command "STATISTICAL ANALYSIS"

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U.S. JEPARTMENT

OF

COMMERCE

National

Bureau

of Standards
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A Users' Guide to the OMNITAB

Command "STATISTICAL ANALYSIS"

H. H. Ku

Institute for Basic Standards

National Bureau of Standards Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of

transient or limited interest. Technical Notes may be

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U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, D/'recfor

Issued March 1973
National Bureau of Standards Technical Note 756

Nat. Bur. Stand. (U.S.), Tech. Note 756, 4 4 pages (Mar. 1973) CODEN: NBTNAE

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

Price $0.75 domestic postpaid or $0.50 G. P.O. Bookstore.
TABLE OF CONTENTS Page

Introduction

STATISTICAL ANALYSIS 4

I. The Command 4

II. General Comments 6

Frequency Distribution 7

Measure of Location 7

Measure of Dispersion 10

Linear Trend Statistics 15

Tests for Non- randomness 16

Other Statistics 19

Ranked Observations 21

Ordered Observations 21

References 23

Appendix A - Tables 24

Appendix B - An Example of Output 32

III

A USERS' GUIDE TO THE OMNITAB COMMAND "STATISTICAL ANALYSIS"

H. H. Ku

Institute for Basic Standards

Abstract

This Technical Note is the first of a series of interpretive notes for a number of

commands in the OMNITAB system that have the automatic printout feature. Others planned

in the series include commands FIT and POLYFIT, TWOWAY analysis, ONEWAY analysis, and

CORRELATION.

These notes aim to be self-contained so that users may have sufficient information

on hand for the understanding of the statistics computed and to use them for their

immediate applications. . Computation formulas are given in the text, and a number of

statistical tables are reproduced in the Appendix for the convenience of users.

Key Words: Computing, statistical; documentation for users; OMNITAB; statistics.
Introduction

The purpose of this Technical Note is to give some supplemental statistical information for the OMNITAB command STATISTICAL ANALYSIS described in NBS Handbook 101 and NBS

Technical Note 552.— This command (and several others) gives automatic printout of a

variety of statistics computed from a set of observations. The interpretations of these

statistics and their uses are described in some detail in this Note.

This Technical Note aims to be self-contained in the sense that users may gain

sufficient understanding from the descriptions of these statistics for their immediate

applications. References to popular statistical texts are given whenever available and

to papers in statistical literature only when discussions are not readily found in text

books. Computation formulas are provided. A number of statistical tables referenced are

reproduced in the Appendix for the convenience of the user.

The printout for STATISTICAL ANALYSIS used in this Note is based on OMNITAB II,

version 5.01, dated May 29, 1971. Future changes in printout format and algorithms,

however, would not vitiate the usefulness of the descriptions and interpretations given

for these statistics in this Note.

We assume that readers already know how to use OMNITAB and are familiar with the

OMNITAB User's Reference Manual. We only note a few related items of interest.

A. The command STATISTICAL ANALYSIS works on a column of numbers. These

numbers may be read into the desired column by the commands READ or SET,

or may be stored in a column as a result of another command, e.g., residuals

stored as a result of the command FIT.

B. The normal size of the worksheet is 201 rows by 62 columns. If the length

of the data exceeds 201, then the arrangement of the worksheet has to be

changed by the command DIMENSION.

1/ OMNITAB: A Computer Program for Statistical and Numerical Analysis, Nat'l. Bur. of Standards Handbook 101 (1966, reissued 1968).

OMNITAB II User's Reference Manual, NBS Technical Note 552 (1971).
C. If STATISTICAL ANALYSIS is to be used repeatedly for several columns of

different lengths, the command RESET must be used to set the length of

each column.

The first part of the following lists the command STATISTICAL ANALYSIS in its

various forms, and the second part describes and interprets statistics given in the

printout.

Frequently users may wish to use weights in conjunction with their observations.

The command "STATISTICAL ANALYSIS" can be used with a specified column of weights. The

use of such weights, however, raises problems in the strict interpretation of some of the

statistics that are printed automatically. We shall not attempt to interpret all such

items, except to give the rule of computation used in the weighted cases:

a. If the weights are either O's or l's, the observations with zero weights

are excluded from computation. The number of observations, N, is taken to

be those with weights one.

b. If arbitrary weights, other than O's and l's, are given, then all the

deviations of the observation are computed from the weighted mean, i.e.,

at all places where the term "Dev." is used,

DEV = OBSERVATION - WEIGHTED MEAN.

These "deviations" are used in the formula given in the text, with or

without weights, regardless as to whether the resulting number is

meaningful.
STATISTICAL ANALYS I

S

I. The Command 1/

The simplest form of the command for statistical analysis is:

STATIStical analysis of values in column (C)

or, in an abbreviated form

STATIS (C)

At least two pages of automatic printout will result from this command, as shown in the

example in Appendix B. The first page lists values of 53 statistics computed from the

observations in Column C, and the second page lists, among others, ranks of these observations and ordered observations. A detailed discussion on the use of these statistics is

given in the next section under GENERAL COMMENTS.

If storage of these statistics in specified columns is desired for later use, e.g.,

plotting of deviations from the mean, the command can be modified by adding either 1 or 4

arguments as follows:

STATIStical analysis of column 2, put statistics in column 10

(and next three columns)

STATIStical analysis of column 2, put statistics in columns 10, 11, 12, 13

The results of the above two commands are in fact equivalent. The second form, however,

can be used to specify the storage in specified columns that are not in a sequence.

In the above example, values of the 53 statistics will be stored in the rows of- column

10 as shown in Appendix B, the ranks of the observations in column 11, the ordered

observations in column 12, and the residuals, or deviations from the mean, in column 13.

In the three forms of the command given above, all weights are set equal to ,1. If

a column of weights is used, then the number of arguments must be increased to include

2/ For a detailed description of this command, see pp. 115-122, OMNITAB II USer's Reference Manual, NBS Technical Note 552, issued October 1971.
the column of weights as follows

STATIStical analysis of column 2, weights in column 3, put statistics

in no columns -4

STATIStical analysis of column 2, weights in column 3, put statistics

in column 10 (and next three columns)

STATIStical analysis of column 2, weights in column 3, put statistics

in columns 10, 11, 12, 13

A summary of the number of arguments in various forms of this command is given below.

NUMBER OF ARGUMENTS IN COMMAND STATIS

No Storage Consecutive Storage Specified Storage

No weights (weights=l) 1 2 5

Weights specified -3* 3 6

* Last column number is preceded by a minus sign.

In each form of STATIS described above, an "S" in front of the command (SSTATIS)

suppresses the automatic printout.

The example in Appendix B used the following commands

OMNITAB DAVIS-HARRISON RELATIVE HUMIDITY DATA PIKES PEAK POSITIVE DEW POINT

READ 1, 2

84 data points

STATIS 2, 10, 11, 12, 13

PRINT 10, 11, 12, 13

STOP

(Identification numbers of data points are read into column 1.)
II. General Comments

Given a column of N numbers, what can we say about them? Evidently the answer depends

on the information we are looking for, and different people would ask different questions.

The command "Statistical analysis" is designed to answer almost all the questions that

might be raised and can be answered - a 99.44% extractor.

Hence at least two pages of statistics are printed out automatically for each statistical analysis of a column of N numbers (unless printout is suppressed) . Some of these

statistics, such as the average, standard deviation, are nearly always required by the

user; others may provide information which he did not ask for. We therefore feel that it

is to the advantage of the user to make a practice to glance over the sheets and look for

surprises that may contradict his general feeling or impression about the data. In this

sense additional information may be gained through the use of this command.

For convenience, the statistics computed are classified into eight groups:

1. Frequency distribution

2. Measures of location

3. Measures of dispersion

4. Linear trend statistics

5. Tests for non-randomness

6. Other statistics

7. Ranked observations

8. Ordered observations

We will give a brief general discussion of the statistics given within each group, and the

uses they may be put to. Obviously these discussions cannot be exhaustive and references

are given whenever appropriate, in addition to those given in the printout in parenthesis.

We shall use the attached printout as an example for our discussion.

OMNITAB DAVIS-HARRISON R.H. DATA, PIKES PEAK POS. P.P. LOW

The first line of the printout, as Is always the case for OMNITAB for every page,

is used for name, date, title of the experiment, or any other labels of identification.
STATISTICAL ANALYSIS OF COL 2 N=84

The second line gives the name of the command and the number of data points,

N=NRMAX. This is useful to check for duplication or omission of cards.

FREQUENCY DISTRIBUTION (1-6) 5 25 35 8100442

The next line gives the frequency distribution which shows the number of data points

that fall in each of 10 equal intervals of the range of all the data, listed in ascending

sequence of intervals. Items in parentheses refer to page number in NBS Handbook 91.

The frequency distribution is helpful in:

(1) looking at the shape of distribution, e.g., skewness, peakedness, etc., and in

(2) detecting outliers at one or both ends.

Usually if an end point is separated by one or more zeroes from the main part of the

data in the frequency distribution, it is suspected as an outlier. If it can be verified

that the outlier is present because of a mistake in recording or typing, or because of

violation of experimental conditions, obviously the point should be deleted. Other than

that the course of action is not clear. We suggest as a first reference the three-page,

"Some Remarks on Wild Observations" by William H. Kruskal [1]

The data shown are a partial set of calibration data for a whirling psychrometer made

by William Ferrel in 1886 on Pikes Peak, Colorado. There appear to be 10 high values

separated by two zero class intervals from the remaining 74 points. A close look at these

points is in order before we proceed to further analysis.

MEASURES OF LOCATION (2-2)

UNWEIGHTED MEAN = 6.3734391-01

WEIGHTED MEAN = 6.3734391-01

MEDIAN = 6.2915000-01

MID- RANGE = 6.6844999-01

25 PCT UNWTD TRIMMED MEAN = 6.2885945-01

25 PCT WTD TRIMMED MEAN = 6.2885945-01

For a description of these six statistics all of which are measures of location, we

first introduce a column of weights, and let W. stand for the weight for the ith row cor-
responding to the number X.. For later use, we also denote the number of non-zero weights

as "NZW".

The weighted mean is therefore:

N

I (W.X.) WEIGHTED MEAN

N

y W. 1=1 X

If all W. = 1 (or W. = k) then we obtain the unweighted mean

1 N i. I x - N . L . 1 1=1

For the weighted mean, the user must have reason to select the weights he wishes to

attach to the X values. Some of the common ones could be:

1. To assign a weight of zero to a value to be omitted in the average,

2. To assign weights proportional to the number of observations which the X. are

averages of, or

3. To assign weights inversely proportional to variances of individual X. values, provided these variances are fairly well established.

From the expression for weighted mean above, it is evident that the computation is

carried out in a self-normalizing manner, and the sum of weights need not be 1.0 in

the input.

We may continue our discussion on the median, the midrange, and the 25% trimmed mean

as special cases of weighted mean, although the weights used are incorporated in the

computing routine and do not need to be specified. N+l For N odd, set W. = 1 for i = —r— , W. = for all other i: we obtain the median,

l 2 i N N For N even, the value of the median is computed with W. = 1 for i = —, i = — + 1, and

zero otherwise.

The midrange is defined as the average of the two extreme values, hence the weights

are l's for the smallest and largest values in the column of values of X, and 0's for the

remaining values.
The selection of these three measures of location is not without justification. To

begin with, each measure represents the solution of a particular minimization criterion.

If one wishes to choose the measure of location, m, such that

N

l (X. - m) = min , UNWEIGHTED MEAN is the resulting estimate of m, i=l x

N

I |X - ml = min, MEDIAN is the result,

i=l X

maxlx - ml = min, MIDRANGE is the result.

. 1 1

Furthermore, each measure is the "best" estimator of the location parameter for

certain distribution from which the observed values were obtained (as a random sample)

Heuristically the "best" estimator may be interpreted as the most "stable" estimator for

the parameter in repeated samples from the same population. Thus the unweighted mean is

the "best" if these values are normally distributed, the median is the "best" if the

distribution is double exponential, and the midrange the best if rectangular.

Hence if we know approximately how our data are distributed, we can select the proper

measure of location. Most of the time, however, we do not enjoy this privilege, and wish

that we could use an estimate that is "robust" for a reasonable number of likely distributions. A "robust" estimator is particularly important if the data may be contaminated

by outliers, or by a cluster of points originating from another distribution.

The 25 percent trimmed mean is such a robust estimator. Both the upper and lower 25%

of the ordered values are "trimmed" off, and the mean is computed from the middle 50% of

the values. In our example, 21 values from each end are assigned weight of 0's, and the

mean is computed from the central 42 values all of which fall within the second and

third intervals of the frequency distribution. We note that the ten high values do not

contribute at all to the. 25% trimmed mean. We note also that the value of the median,

itself an extreme trimmed mean, is very close to the value of the 25% trimmed mean.

What it amounts to is that we are paying an insurance premium, half of the observed

values, to minimize the chance that one or more aberrant values may unduly disturb the

mean. Also, parallel to insurance premium payments, the protection is for the long run,
since for just one series of observations we do not know if the "trimming" is worth the

cost.

Measures of location is a large subject in itself. The measures of location computed

and printed out in STATISTICAL ANALYSIS offer one a chance to inspect and compare these

measures, as well as a choice of one or another measure for the class of unimodal continuous distributions considered (distributions with a single peak). For a recent discussion we recommend Crow and Siddiqui [2].

MEASURES OF DISPERSION (2-6)

STANDARD DEVIATION = 3. 2401807-02

S. D. OF MEAN = 3.5353269-03

RANGE = 1.4670000-01

MEAN DEVIATION = 2.1074478-03

VARIANCE = 1.0498771-03

COEFFICIENT OF VARIATION = 5.0838811-00

As in the case of measures of location, we have a column of N values of X., and a

column of weights W.. If the weights are not specified, W. = 1 for the computed results.

Zero weights may be specified to delete one or more numbers. Here we are interested in

some measure of the scatter of values of X..

l RANGE

The RANGE of N values is defined as the difference between the highest and the

lowest of these values, i.e. X - X . . If we have only two values of X, the difference max mm

between the two numbers is all the information we have about the dispersion. The same

is reasonably true for small sets of numbers. However, as N gets larger, the range is no

longer an efficient measure of dispersion. Evidently, the range is not affected by

weights.

VARIANCE

The most important measure of dispersion, the VARIANCE computed from the set of N

values, is

, N _ N

S

Z =

I (X. - X)

Z /(N-1) =

I (DEV.)

Z /(N-1) i=l 1 i=l 1

10
or, if weights are used, is

? N „ S =

I (X. - WEIGHTED MEAN) W./(NZW - 1) i=l 1 X

N

2 =

I (DEV.) W./(NZW - 1), i=l x X

where the weighted mean is computed as

N N

y w.x. / y w. . i=i i=i

We shall return to the weighted case later

2 There are a number of reasons for S to be the most popular measure of dispersion,

2 2

1. S is an efficient and unbiased estimator of a , the variance.

2

2. S is additive for independent variables or observations, hence its use in

propagation of errors, and pooling of estimates of variances.

2

3. S is mathematically more tractable in comparison with other measures.

2 It is impossible to discuss even a fraction of the properties of uses of S in the

limited space here. We suggest as a start, in addition to the section of NBS Handbook 91

given in parentheses, that one look over the 15 items listed under variance and those

under standard deviation in the index of NBS Special Publication 300, Vol 1. , reference

[3].

STANDARD DEVIATION

The positive square root of the variance computed from N numbers is the computed

STANDARD DEVIATION, S in symbol. S is of the same unit as the observed values and is

usually used as the measure of precision of these values. Two comments here are

appropriate:

1. The standard deviation computed is the standard deviation of X. about the mean X, and

is referred to as the "standard deviation of a single observation."

With increasing N, S approaches a constant value which is the standard deviation

of the population, or of the measurement process.

11
2. The standard deviation S is only an estimate computed from N numbers, and its

"firmness" depends entirely on N. It is incomplete to report the value of S without

also indicating in some manner the value of N from which S is computed.

STANDARD DEVIATION OF MEAN (Also called Standard Error of the Mean)

The S. D. OF MEAN is computed as

S. D. OF MEAN = S / /n

or, in the case of the weighted mean

S. D. OF WEIGHTED MEAN = S / Aw.

We note that, contrary to S which approaches a constant value with increasing N, the

S. D. of mean approaches zero with increasing N. The number of measurements, N, is therefore an important item to report together with the mean and the standard deviation of the

mean.

WEIGHTED CASES FOR VARIANCE AND STANDARD DEVIATION

Whenever a column of weights is specified, then the STATIS command works only on those

numbers with non-zero weights , and this fact is stated in the third line of the printout.

Hence, if the column of weights consists of l's and O's only, the weighted and unweighted

statistics appearing in the printout are the same.

The main reason for using weights is to combine observations having different precisions. A convenient way to think about this problem may be described as follows:

Suppose each X. is actually an average of a number of observations, say N.. If the

numbers N. for the groups are different from group to group, then a weighted average with

the weights proportional to N. is a proper mean to compute, such that the weighted average

is equal to the simple average of the original data set. Assuming that there is no

between group variation and that the variance for the individual observations are the same

2 and equal to a , or assuming

VAR(X.) = a.

2 = a

2 /N. ,

2 we select W. to be inversely proportional to a., or proportional to N. . Hence we can

think of the column of weights as the number of observations that enter into each of the

averages X.

12
One can show that the variance computed by

2 N2 S = I (DEV.) N./(NZW - 1) i=l 1 1

2 appearing in the printout is an unbiased estimate of a , the underlying common variance

for an individual observation. The standard deviation appearing in the printout corresponds also to this interpretation.

2 To obtain the estimated variance of a particular X., we will have to divide S by N.

l

J l i.e. ,

VAR(X.) = S

2 /N. , or S

2 /W.

x 1 1 and

S.D.(X.) = S / /N. , or S / /W. . i l l

The S. D. of mean for the weighted case is the standard deviation of the weighted

mean computed by

S. D. OF MEAN = S / /iW.

i

X

COEFFICIENT OF VARIATION

The coefficient of variation appearing in the print-out is expressed in percent and

is defined as

COEFFICIENT OF VARIATION = 100(S/X).

Sometimes the term "relative standard deviation" is used for the same expression. For the

weighted case, the weighted mean is used in the denominator.

The coefficient of variation is a constant when magnitudes of random errors are proportional to averages of the observations, and hence is a useful measure of dispersion to

summarize data when the same coefficient of variation applies to a set of different means.

Its use should be restricted to this type of data only.

Since S is positive by definition, the coefficient of variation computed will be negative if X is negative. The use of coefficient of variation is therefore meaningful only

when all the observations take positive values, or if X is bounded away from zero.

13
MEAN DEVIATION

N MEAN DEVIATION = Y JX. - xl/N , 1=1 X

where X is the weighted mean if weights are specified.

The mean deviation has been popular with physical scientists for a long time: it

is easy to compute for small N and it is less affected by outliers than the standard

deviation. If the observations are exactly normally distributed, then the computed

standard deviation S is a more efficient measure of dispersion than the mean deviation.

But if the observations are not normally distributed, or are contaminated by outliers,

then the advantage of the estimator S over that of the mean deviation is reduced rapidly.

For a general discussion of efficiency of the mean deviation as compared to other

measures of dispersions in contaminated distributions see Tukey (Ref. A).

A convenient way to check the normality of one's data is to remember that the ratio

of mean deviation to standard deviation is about 0.8 for large samples from normally

distributed values. Percentage points of the distribution of this ratio is given in

Table V which can be used to test for departure from normality.

For data that are normally distributed, an approximate relationship between the mean

deviation and the standard deviation computed is as follows

STD. DEVIATION = MEAN DEVIATION x £ x -^r

2 N-l

Hence a comparison of the standard deviation computed from data and that computed from

the expression above provides another check on the normality assumption of the data.

A TWO-SIDED 95 PCT CONFIDENCE INTERVAL FOR MEAN IS

6.3031-01 TO 6.4438-01 (2-2)

A TWO-SIDED 95 PCT CONFIDENCE INTERVAL FOR S.D. IS

2.8026-02 TO 3.8586-02 (2-7)

The next two lines of printout give a two-sided 95% confidence interval for the mean

u, and a two-sided 95% confidence interval for the standard deviation a, assuming the

observations are normally distributed. The upper and lower limits are computed by the

14
following formulas:

upper limits lower limits

u x

u

= X + tS / /N x

l

= X - tS / /i

aS=BS u u S=BS °L L

where t (a = 0.05, n-1) is the Student's t-f actor given in Table II, and

B , B (a = 0.05, n-1) are factors for two-sided confidence limits for a given in Table III

in the appendix.

Since interval estimation is a large subject in itself, we refer to Chapters 1 and 2

of reference [6] for the concepts and interpretations of these intervals. Suffiae

it to note here that the confidence level used, 95% in these cases, is associated with

the procedure used in the computation such that 95% of the intervals so computed would

bracket the mean u (or standard deviation a) in repeated sets of observations that are

normally distributed. For a particular set of data, the intervals so computed may or may

not bracket the mean (or standard deviation)

LINEAR TREND STATISTICS (5-1)

SLOPE = -2.4760868-04

S.D. OF SLOPE = 1.4412009-04

SLOPE/S.D. OF SLOPE = T = -1.7180820+00

PROB EXCEEDING ABS VALUE OF OBS T = .090

A set of data is always taken in some sequence, usually ordered in time, or by entry

of data from the design. The linear trend statistics check on the drift, if any, of the

data against the order of the sequence. Essentially the data are plotted against the

integers 1, . .

. , N, and a line is fitted by the method of least squares. The computation

formulas for the slope and the standard deviation of the slope are the following:

15
Slope = 12 I i(DEV.)/N(N - 1) i=l 1

S.D. of slope

/N-2

12 I (DEV.)' 1=1

(SLOPE)'

N(N -1)

1/2

If the assumption is no trend in the sequence, then

Slope - 0.0

S.D. of slope

is distributed as Student's t with (N-2) degrees of freedom. The probability of observing

a slope of the magnitude calculated or greater while in fact there is no trend is given in

the next line. In this example the probability is only .090, giving some evidence against

the assumption of no trend (including but not necessarily limited to linear trend) . A

plot of the data or residuals becomes almost mandatory at this juncture of analysis: to

reveal possible causes of the trend. (In searching for trend when the data are entered in

some meaningful order, we should bear in mind that the trend in the data may also be

caused by chance with the probability indicated)

.

TESTS FOR NON- RANDOMNESS

NO. OF RUNS UP AND DOWN

EXPECTED NO OF RUNS

S.D. OF NO OF RUNS MEAN SQ SUCCESSIVE DIFF MEAN SQ SUCC DIFF/VAR

DEVIATIONS FROM WTD MEAN

NO OF + SIGNS

NO OF - SIGNS

NO OF RUNS EXPECTED NO OF RUNS

S.D. OF RUNS

DIFF. /S.D. OF RUNS

4755.7 3.82 3.6380990-04

.347

22621433.5 3.51 -5.550

16
Since "randomness" is an elusive and difficult concept to define and test, our tests

here are restricted to the detection of certain important patterns or non-randomness

that may be inherent in a set of data which is ordered in some meaningful way. In the

linear trend statistics described above, the test is for long-term trend from the beginning

to the end, and is based on normal theory; in this section the tests are for the detection

of short-term patterns, such as grouping of data into sets, cyclic variations, and the

like. Moreover, the tests used do not depend on any assumption on the form of distribution.

Let us take our sequence of observations: X., X„, ..., X and look at the algebraic

signs of the differences between adjacent numbers, i.e., the signs of (X. - X.). There

are (N-l) such differences, giving rise to a sequence of (N-l) "+" or "-" signs. A subsequence of successive "+" signs is called a run up and a subsequence of "-" signs is

called a run down. Assuming that all N! possible arrangements of the N numbers are equally

likely, then it can be shown that:

EXPECTED NO. OF RUNS = -|(2N-1), and

S.D. OF NO. OF RUNS = [ (16N-29) /90

1' 2

.

Furthermore, for N > 20, the number of runs may be regarded as normally distributed and

Table I in the Appendix can be used to check if the departure from expected is excessive.

(See for instance, p. 354, Reference [7].) Hence the ratio:

OBSERVED NO. OF RUNS - EXPECTED NO OF RUNS

S.D. OF NO. OF RUNS

gives a measure of the non- randomness of the sequence. Short-term drifts would tend to

yield a smaller number of runs than expected. Cyclic data could yield either larger or

smaller number of runs, depending on the period of the cycle.

The mean square successive difference is another test for the presence of correlation

between adjacent observed values, and is defined as:

N

2 MEAN SQ SUCC DIFF =

J (X - X.) /(N-l) . i=l The ratio:

MEAN SQ SUCC DIFF/VAR

appearing on the next line has an expected value of 2. To test if there are correlations

17
between adjacent values, it is simpler to consider r = — (MEAN SQ SUCC. DIFF./VAR) which

has an expected value of 1 and standard deviation ofJ(N-2)/(N -1). For N > 20 this ratio

is approximately normally distributed. Hald [7], gives a table of percentage points for

N < 20 for this ratio which is reproduced as Table IV in the appendix. A ratio significantly less than one is an indication of positive correlation between adjacent values,

and a ratio greater than one that of negative correlation.

Hald (page 358, reference [7]) illustrates the use of this statistic for checking

gradual changing of the mean of the population while the variance remains constant. The

2 2 2 estimate S of a will tend to be too large because S includes also variations of the

population mean. On the other hand, since the mean square successive difference includes

only the differences between successive values, the effect of gradual changes in the

population mean will be relatively small.

If we compute the values of (X. - X) and count the number of "+" and "-" signs, we

obtain the first two lines under DEVIATIONS FROM WTD MEAN:

NO OF + SIGNS = M

NO OF - SIGNS = M

(If a value of X. is an integer and is equal to X, a "+" sign is counted, hence

M. + M = N always). In the weighted case, the weighted mean is subtracted from the

observations.

Each change in the sequence of signs constitute a run. The total number of runs

in a sequence is equal to the number of change of signs plus one, and is given in the

next line. Again assuming all arrangement of "+" and "-" signs are equally likely, then

it can be shown that (p. 170, reference [8]):

2M M

EXPECTED NO OF RUNS = 1 + —~~-

S.D. OF RUNS = 2M M (2M M - N)

N

2 (N-1)

1/2
The distribution of NO OF RUNS becomes approximately normal for large samples, and is

usually good enough for practical purposes for M , M > 10. The last line of this

section gives

NO OF RUNS - EXPECTED NO OF RUNS

S.D. OF RUNS

which can be used to check for abnormally large departure of the number of runs from the

expected. Table I in the appendix can be used for this purpose.

OTHER STATISTICS

MINIMUM

MAXIMUM

BETA ONE BETA TWO WTD SUM OF VALUES WTD SUM OF SQUARES

WTD SUM OF DEVS SQUARED

STUDENT'S T WTD SUM ABSOLUTE VALUES

WTD AVE ABSOLUTE VALUES

5.9509999-01

7.4180000-01 3.7310967+00

5.9296834+00

5.3536888+01 3.4208556+01

8.7139798-02 1.8027863+02

5.3536888+01

6.3734391-01

The smallest and largest values in the set of data are printed out as the minimum

and maximum, respectively, as the first two lines in OTHER STATISTICS.

The two statistics BETA ONE and BETA TWO are related to the third and fourth

moments of the distribution. They are included here to give an indication of the

normality of the observed data, and are calculated by the following formulas:

BETA ONE = b, N i.VDEV

3 1=1

N if (BEV.)

2 i=l

, estimating the value of 8 for the population;

BETA TWO =b 2 = f

I(DEV.)

N

I i=l

1 N9 i I (»EV.)

2 1=1

, estimating the value of B„ for the population.

Weights, if used, are ignored in these computations except for the calculation of (DEV.),

19
The value b. is a measure of asymmetry (skewness) of the distribution of observed

values, and the value b„ is a measure of its "peakedness" (kurtosis). In a normal

population, 3, = 0, and B- = 3. For B, > 0, the distribution has a longer tail

than the normal distribution. For 8

? > 3, the distribution has a sharper peak

than normal; for B„ < 3, a flatter peak than normal. For uniform distribution, g. is

equal to 1.8. However, since these two statistics fluctuate considerably from sample to

sample, they are useful only for moderately large groups of data. Percentage points of

the distributions of /b. and b„ are given in Table Vb and Table Vc respectively.

Tables and discussions of tests of skewness and kurtosis are given in reference [9].

Since a test for normality can be considered simultaneously as a test for the

presence of outliers, these statistics have also been proposed for such tests, especially

when more than one spurious observation may be involved. For a discussion see reference

[10].

If the observed values are believed to center about zero, the STUDENT'S t statistic

can be used to check this assumption. It is computed as:

WTD MEAN - 0.0 .63734391-0 STUDENT'S T = S.D. OF MEAN .0035353269

Percentiles of the "t" distribution are given in Table II in the appendix.

The remaining five items in OTHER STATISTICS are self-explanatory. These are computed as follows:

N WTD SUM OF VALUES = 'T W.X. 1=1 X X

N , WTD SUM OF SQUARES = £ W.XT

i=l

N , WTD SUM OF DEVS SQUARED = Y (DEV.) W. i=l X X

20
WTD SUM ABSOLUTE VALUES = V Iw.X. i=l

1 N WTD AVE ABSOLUTE VALUES = t ) W.X.I . N . L ' 1 l

N

I1=1

The above completes the descriptions of the various statistics appearing on the first

page of the printout.

RANKED OBSERVATIONS

I X(I) RANK X(I) - MEAN

ORDERED OBSERVATIONS

NO. X(J) X(J+1) - X(J)

Two blocks of information are presented on the next page of printout, with titles

"OBSERVATIONS" and "ORDERED OBSERVATIONS" appearing on top of the left and right groups

of columns respectively. There are five, or four, columns under "OBSERVATIONS",

depending on whether weights are used or not used. The first column "I" indexes the

observations in the sequence they are read into the program. The second column X(I) lists

the values of observations in floating point format. The third column lists the ranks of

the values when ranked from smallest to largest. If two or more values are the same,

these are given in the same rank number, calculated as the average of the ranks if these

numbers are slightly different. In our example, X(67) and X(80) share the rank 56.5, and

X(23), X(41) and X(78) share the rank 33.0.

The fourth column lists the differences X(I) - MEAN in floating point format. The

weighted mean is used if weights are specified. These weights are printed as W(I) in the

fifth column, only when weights are specified.

Since there are ten high values spearated from the main body of data in our example

(as shown in FREQUENCY DISTRIBUTION) , we certainly would like to know what these values

are. From the column of ranks we find that ten consecutive values X(ll) thru X(20) have

consecutive ranks 75 thru 84, with positive X(I) - MEAN values approximately an order of

magnitude larger than the others. Mr. Harrison checked into the original data and found

a note to the effect that these values were obtained in a heated cabin, whereas the other

values were obtained in open air.

21
An easier way to spot these ten values is to look at the second column X(J) in the

next block under the title ORDERED OBSERVATIONS. The column X(J) orders the observations

in an ascending sequence, hence the ten largest values appear at the end of the column.

Under the heading "NO." the index number "I" associated with the observations are listed

so that specific ones can be identified. The last column X(J+1) - X(J) gives the successive differences between adjacent ordered observations.

From the apparent grouping of adjacent observations (e.g. X(l) through X(10) are all

at the low end, etc.) of this set of data, it seems that there is considerable

heterogeneity (and perhaps a between day component) which should be taken into consideration in further analysis.

COLUMN 10 COLUMN 11

84.000000 3.0000000

COLUMN 12

.595099999

COLUMN 13

-.030343913

The command we used for this example

STATISTICAL 2, 10, 11, 12, 13

stored the results of computations in columns 10, 11, 12, and 13. These columns are

printed on the next page for ready reference. The 43 statistics plus the 10 values of the

histogram appear in column 10 in the order listed on page 119 of the OMNITAB II User's

Reference Manual. The ranks of these values are printed as column 11; the ordered values

in column 12; and the deviations (residuals) in column 13. These values can be extracted

from the respective columns for use in subsequent computations. For example, the residuals

can be plotted against the order of observations; the mean and standard deviations may be

"DEFINED" to rows of other columns as elements of a table where several sets of data are

summarized; etc. since it is impossible to list even a small fraction of what one can do

with these stored values, we refer the user to the "self-teaching" feature of OMNITAB to

experiment to his satisfaction that OMNITAB is indeed an easy and effective system to use

in statistical or data analysis. Self-teaching is discussed on pages 39-40 of the OMNITAB

II User's Reference Manual.

22
REFERENCES

[1] Kruskal, William H. , Some remarks on wild observations. Technometrics . 2, No. 1, 1-3 (1960). Reprinted as a paper in 5.5 in Reference [3] below.

[2] Crow, Edwin L. and Siddiqui, M. M. , Robust estimation of location. Journal of the American Statistical Association , Vol. 62, pp. 353-389, 1967.

[3] Ku, Harry H. , Ed. , Precision Measurement and Calibration - Statistical Concepts and Procedures . National Bureau of Standards Special Publication 300, Vol. 1, 1969.

[4] Tukey, John W. , A survey of sampling from contaminated distributions. Contributions

to Probability and Statistics . Ingram Olkin, Editor, Stanford University Press, 1960.

[5] Pearson, E. S. and Hartley, H. 0., Biometrika Tables for Statisticians , Vol. 1, Cambridge University Press, 2nd edition, 1966.

[6] Natrella, Mary G. , Experimental Statistics . National Bureau of Standards Handbook 91, 1963, 2nd edition, 1966.

[7] Hald, A., Statistical Theory with 'Engineering Application . John Wiley and Sons, Inc.,

1952.

[8] Brownlee, K. A. , Statistical Theory and Methodology in Science and Engineering . John Wiley and Sons, Inc., 1960.

[9] Snedecor, George W. and Cochran, W. G. , Statistical Methods . 6th edition, The Iowa State University Press, 1967.

[10] Grubbs , Frank E. , Procedures for detecting outlying observations in samples.

Technometrics , Vol. 11, No. 1, 1969.

23
APPENDIX A

TABLE I (A-l Ref. 6). Cumulative Normal Distribution

TABLE II (A-4 Ref. 6). Percentiles of the t Distribution

TABLE III (A-20 Ref. 6). Factors for Computing Two-Sided Confidence Limits for a

2 2 TABLE IV (13.14 Ref. 7). Fractiles in the Distribution of r = q /s

(r = MEAN SQ SUCC DIFF/2 VAR)

TABLE V (34 Ref. 5). Tests for Departure from Normality

A. Percentage points of the distribution of

a = (mean deviation) / (standard deviation) — q /o B. Percentage points of the distribution of /b 1 = m /m

2 C. Percentage points of the distribution of b. = m,/m

24
TABLE I. CUMULATIVE NORMAL DISTRIBUTION — VALUES OF P -P :!_

Values of P corresponding to Zp for the normal curve.

z is the standard normal variable. The value of P for — ip equals one minus the value of P for +ip,

e.g., the P for -1.62 equals 1 — .9474 = .0526.

ZP

.00 .01 .02 .03 .04 .05 .06 .07 .08 .09

.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359

.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753

.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141

.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517

.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879

.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

1.0 .8413 .8438 .8461 .8485. .8508 .8531 .8554 .8577 .8599 .8621

1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830

1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015

1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177

1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817

2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857

2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890

2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916

2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936

2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952

2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964

2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974

2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981

2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986

3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990

3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993

3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995

3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997

3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

Adapted from Experimental Statistics by Mary G. Natrella, 1963, NBS Hand- book 91, U. S. Government Printing Office, Washington, D. C. , 20015.

25
TABLE II. PERCENTILES OF THE t DISTRIBUTION

df 1.60 £.70 t .80 *.90 *95 £.975 *99 *.995

1 .325 .727 1.376 3.078 6.314 12.706 31.821 63.657

2 .289 .617 1.061 1.886 2.920 4.303 6.965 9.925

3 .277 .584 .978 1.638 2.353 3.182 4.541 5.841

4 .271 .569 .941 1.533 2.132 2.776 3.747 4.604

5 .267 .559 .920 1.476 2.015 2.571 3.365 4.032

6 .265 .553 .906 1.440 1.943 2.447 3.143 3.707

7 .263 .549 .896 1.415 1.895 2.365 2.998 3.499

8 .262 .546 .889 1.397 1.860 2.306 2.896 3.355

9 .261 .543 .883 1.383 1.833 2.262 2.821 3.250

10 .260 .542 .879 1.372 1.812 2.228 2.764 3.169

11 .260 .540 .876 1.363 1.796 2.201 2.718 3.106

12 .259 .539 .873 1.356 1.782 2.179 2.681 3.055

13 .259 .538 .870 1.350 1.771 2.160 2.650 3.012

14 .258 .537 .868 1.345 1.761 2.145 2.624 2.977

15 .258 .536 .866 1.341 1.753 2.131 2.602 2.947

16 .258 .535 .865 1.337 1.746 2.120 2.583 2.921

17 .257 .534 .863 1.333 1.740 2.110 2.567 2.898

18 .257 .534 .862 1.330 1.734 2.101 2.552 2.878

19 .257 .533 .861 1.328 1.729 2.093 2.539 2.861 20 .257 .533 .860 1.325 1.725 2.086 2.528 2.845

21 .257 .532 .859 1.323 1.721 2.080 2.518 2.831 22 .256 .532 .858 1.321 1.717 2.074 2.508 2.819

23 .256 .532 .858 1.319 1.714 2.069 2.500 2.807 24 .256 .531 .857 1.318 1.711 2.064 2.492 2.797 25 .256 .531 .856 1.316 1.708 2.060 2.485 2.787

26 .256 .531 .856 1.315 1.706 2.056 2.479 2.779

27 .256 .531 .855 1.314 1.703 2.052 2.473 2.771 28 .256 .530 .855 1.313 1.701 2.048 2.467 2.763 29 .256 .530 .854 1.311 1.699 2.045 2.462 2.756 30 .256 .530 .854 1.310 1.697 2.042 2.457 2.750

40 .255 .529 .851 1.303 1.684 2.021 2.423 2.704 60 .254 .527 .848 1.296 1.671 2.000 2.390 2.660 120 .254 .526 .845 1.289 1.658 1.980 2.358 2.617

00 .253 .524 .842 1.282 1.645 1.960 2.326 2.576

Adapted by permission from Introduction to Statistical Anu!y.<i< (2d ed.i by W J Dixon and F. J Massoy, Jr., Copyright, 1 9 H 7 , McGr; Company, Inc. Entries originally from Table III of Statistical Tables by ft. A Fish.-r and F. Yatus, 1938. Oliver and Boyd. Ltd., I.o

26
TABLE III. FACTORS FOR COMPUTING TWO-SIDED CONFIDENCE LIMITS FOR

Degrees

of Freedom

* = .05 a = .01 a- .001

" Bu Bl Bu Bl Bu Bl

i 17.79 .3576 86.31 .2969 844.4 .2480

2 4.859 .4581 10.70 .3879 33.29 .3291

3 3.183 .5178 5.449 .4453 11.65 .3824

4 2.567 .5590 3.892 .4865 6.938 .4218

5 2.248 .5899 3.175 .5182 5.085 .4529

6 2.052 .6143 2.764 .5437 4.128 .4784

7 1.918 .6344 2.498 .5650 3.551 .5000

8 1.820 .6513 2.311 .5830 3.167 .5186

9 1.746 .6657 2.173 .5987 2.894 .5348 10 1.686 .6784 2.065 .6125 2.689 .5492

11 1.638 .6896 1.980 .6248 2.530 .5621

12 1.598 .6995 1.909 .6358 2.402 .5738

13 1.564 .7084 1.851 .6458 2.298 .5845

14 1.534 .7166 1.801 .6549 2.210 .5942

15 1.509 .7240 1.758 .6632 2.136 .6032 16 1.486 .7308 1.721 .6710 2.073 .6116

17 1.466 .7372 1.688 .6781 2.017 .6193

IS 1.448 .7430 1.658 .6848 1.968 .6266

19 1.432 .7484 1.632 .6909 1.925 .6333 20 1.417 .7535 1.609 .6968 1.886 .6397

21 1.404 .7582 1.587 .7022 1.851 .6457 22 1.391 .7627 1.568 .7074 1.820 .6514

23 1.380 .7669 1.550 .7122 1.791 .6568 24 1.370 .7709 1.533 .7169 1.765 .6619 25 1.360 .7747 1.518 .7212 1.741 .6668 26 1.351 .7783 1.504 .7253 1.719 .6713 27 1.343 .7817 1.491 .7293 1.698 .6758 28 1.335 .7849 1.479 .7331 1.679 .6800 29 1.327 .7880 1.467 .7367 1.661 .6841 30 1.321 .7909 1.457 .7401 1.645 .6880

31 1.314 .7937 1.447 .7434 1.629 .6917 32 1.308 .7964 1.437 .7467 1.615 .6953 33 1.302 .7990 1.428 .7497 1.601 .6987 34 1.296 .8015 1.420 .7526 1.588 .7020 35 1.291 .8039 1.412 .7554 1.576 .7052 36 1.286 .8062 1.404 .7582 1.564 .7083 37 1.281 .8085 1.397 .7608 1.553 .7113 38 1.277 .8106 1.390 .7633 1.543 .7141 39 1.272 .8126 1.383 .7658 1.533 .7169 40 1.268 .8146 1.377 .7681 1.523 .7197

41 1.264 .8166 1.371 .7705 1.515 .7223 42 1.260 .8184 1.365 .7727 1.506 .7248 43 1.257 .8202 1.360 .7748 1.498 .7273 44 1.253 .8220 1.355 .7769 1.490 .7297 45 1.249 .8237 1.349 .7789 1.482 .7320 46 1.246 .8253 1.345 .7809 1.475 .7342 47 1.243 .8269 1.340 .7828 1.468 .7364 48 1.240 .8285 1.335 .7847 1.462 .7386 49 1.237 .8300 1.331 .7864 1.455 .7407 50 1.234 .8314 1.327 .7882 1.449 .7427

27
TABLE III (Continued). FACTORS FOR COMPUTING TWO-SIDED CONFIDENCE LIMITS FOR a

Degrees

of Freedom

c = .05 * = .01 a = .001

" Bu Bl Bu Bl Bu Bl

SI 1.232 .8329 1.323 .7899 1.443 .7446 52 1.229 .8343 1.319 .7916 1.437 .7466 53 1.226 .8356 1.315 .7932 1.432 .7485 54 1.224 .8370 1.311 .7949 1.426 .7503 55 1.221 .8383 1.308 .7964 1.421 .7521 56 1.219 .8395 1.304 .7979 1.416 .7539 57 1.217 .8408 1.301 .7994 1.411 .7556 58 1.214 .8420 1.298 .8008 1.406 .7573 59 1.212 .8431 1.295 .8022 1.402 .7589 60 1.210 .8443 1.292 .8036 1.397 .7605

61 1.208 .8454 1.289 .8050 1.393 .7621 62 1.206 .8465 1.286 .8063 1.389 .7636 63 1.204 .8475 1.283 .8076 1.385 .7651 64 1.202 .8486 1.280 .8088 1.381 .7666 65 1.200 .8496 1.277 .8101 1.377 .7680 66 1.199 .8506 1.275 .8113 1.374 .7694 67 1.197 .8516 1.272 .8125 1.370 .7708 68 1.195 .8525 1.270 .8137 1.366 .7722 69 1.194 .8535 1.268 .8148 1.363 .7735 70 1.192 .8544 1.265 .8159 1.360 .7749

71 1.190 .8553 1.263 .8170 1.356 .7761 72 1.189 .8562 1.261 .8181 1.353 .7774 73 1.187 .8571 1.259 .8191 1.350 .7787 74 1.186 .8580 1.257 .8202 1.347 .7799 75 1.184 .8588 1.255 .8212 1.344 .7811 76 1.183 .8596 1.253 .8222 1.341 .7822 77 1.182 .8604 1.251 .8232 1.338 .7834 78 1.181 .8612 1.249 .8242 1.336 .7845 79 1.179 .8620 1.247 .8252 1.333 .7856 80 1.178 .8627 1.245 .8261 1.330 .7868

81 1.176 .8635 1.243 .8270 1.328 .7878 82 1.176 .8642 1.241 .8279 1.325 .7889 83 1.174 .8650 1.239 .8288 1.323 .7899 84 1.173 .8657 1.238 .8297 1.320 .7909

85 1.172 .8664 1.236 .8305 1.318 .7920 86 1.171 .8671 1.235 .8314 1.316 .7930 87 1.170 .8678 1.233 .8322 1.313 .7939 88 1.168 .8684 1.231 .8331 1.311 .7949 89 1.167 .8691 1.230 .8338 1.309 .7959 90 1.166 .8697 1.228 .8346 1.307 .7968

91 1.165 .8704 1.227 .8354 1.305 .7977 92 1.164 .8710 1.225 .8362 1.303 .7987 93 1.163 .8716 1.224 .8370 1.301 .7996 94 1.162 .8722 1.222 .8377 1.298 .8004 95 1.161 .8729 1.221 .8385 1.297 .8013 96 1.160 .8734 1.219 .8392 1.295 .8022 97 1.159 .8741 1.218 .8399 1.293 .8031 98 1.158 .8746 1.217 .8406 1.291 .8039 99 1.158 .8752 1.216 .8413 1.289 .8047 100 1.157 .8757 1.214 .8420 1.288 .8055

28
2 2 TABLE IV. Fractiles in the distribution of r = q /s .

Probability in per cent

n 0.1 1.0 5.0

4 0.295 0.313 0.390

5 0.208 0.269 0.410

6 0.182 0.281 0.445

7 0.185 0.307 0.468

8 0.202 0.331 0.491

9 0.221 0.354 0.512

10 0.241 0.376 0.531

11 0.260 0.396 0.548

12 0.278 0.414 0.564

13 0.295 0.431 0.578

14 0.311 0.447 0.591

15 0.327 0.461 0.603

16 0.341 0.475 0.614

17 0.355 0.487 0.624

18 0.368 0.499 0.633

19 0.381 0.510 0.642

20 0.393 0.520 0.650

Adapted with permission from Statistical Theory with Engineering

Applications by A. Hald, 1952, John Wiley and Sons, Inc., New

York.

29
Table V. Tests for departure from normality A. Percentage points of the distribution of a = (mean deviation) /(standard deviation)

Percentage pointa

Size of sample n Mean Standard

deviation n-1

TJpper 1 1%

Jpper 1 5%

Jpper 10%

Lower 10%Lower 5%Lower 1%

11 10 9359 9073 8899 •7409 •7153 •6675 •81805 •05784

16 15 9137 8884 8733 •7452 •7236 •6829 81128 •04976 21 20 9001 8768 8631 •7495 7304 •6950 80792 •04419 26 25 8901 8686 8570 •7530 •7360 •7040 •80590 •04011

31 30 8827 8625 8511 •7559 •7404 •7110 •80456 03697

36 35 8769 8578 8468 7583 •7440 7167 •80360 03447

41 40 8722 8540 8436 •7604 •7470 •7216 •80289 •03241 46 45 8682 8508 8409 •7621 7496 •7256 •80233 '03068

51 50 8648 8481 8385 •7636 •7518 •7291 80188 •02919

61 60 8592 8434 8349 •7662 •7554 •7347 •80122 •02678 71 70 8549 8403 8321 •7683 •7583 •7393 80074 •02487

81 80 8515 8376 8298 •7700 7607 •7430 •80038 •02332

91 90 8484 8353 8279 •7714 7626 •7460 •80010 02203

101 100 8460 8344 8264 •7726 •7644 •7487 •79988 02094

201 200 8322 8229 8178 •7796 •7738 •7629 •79888 01491

301 300 8260 8183 8140 •7828 •7781 •7693 •79855 01220

401 400 8223 8155 8118 •7847 •7807 •7731 •79S38 •01058 501 500 8198 8136 8103 7861 •7825 7757 79828 •00947 601 600 8179 8123 8092 7873 •7838 •7776 •79822 •00865 701 700 8164 8112 8084 7878 7848 •7791 •79817 00801

801 800 8152 8103 8077 •7885 •7857 •7803 •79813 00749

901 900 8142 8096 8071 •7890 7864 7814 •79811 •00707 1001 1000 8134 8090 8066 •7894 •7869 •7822 •79808 •00670

B. Percentage points of the distribution of ^jbx = m^jm\

Size of sample n

Percentage

points Standard

deviation

Size of sample n

Percentage

points Standard

deviation

Size of sample n

Percentage

points Standard

deviation 5% 1% 5% 1% 5% 1%

2530

35

40

45

50

60

70

80

90

100

125

150

175 200

•711 661

•621

•587 •558

•533 492

•459 •432

•409

•389

•350

•321

•298 •280

1-061

•982 921

•869 •825 787

•723

•673

•631 596

•567

•508 •464

•430

•403

•4354

•4052

•3804 3596

3418 3264

•3009

•2806

•2638 2498

2377

2139

•1961 1820

•1706

200 250

300 350 400

450 500

550

600

650 700

750

800

850

900

950

1000

280

251 230

213 200

188

179 171

163 157

151 146 142

138 134

130

127

•403 •360

•329

•305

•285

•269 •255

•243

•233

•224

•215

•208

•202

•196

•190 185

•180

•1706

•1531 •1400 1298

1216

•1147

•1089

•1039

•0995

•0956

•0922

•0891 •0863

•0837 •0814

•0792

•0772

1000

1200

1400

1600 1800 2000

2500

3000

3500 4000

4500

5000

•127 116

107

•100

•095

•090

•080

•073

•068 064

•060

•057

•180 •165 152

•142

•134

•127

•114

•104 096

090

085

•081

•0772

•0705 •0653

•0611 •0576

•0547 0489

0447

•0414 0387

0365

•0346

N.B. As the sampling distribution of -^frj is symmetrical about zero, the same values, with negative

sign, correspond to the lower limits.

Adapted with permission from Biometrika Tables for Statisticians

Vol. 1, edited by E. S. Pearson and H. 0. Hartley, 1958, Cambridge

University Press.

30
Table V. Tests for departure from normality (continued)

C. Percentage points of the distribution of b2 = mjm%

Percentage points Percentage points

Size of sample

Size of sample n Upper Upper Lower Lower n Upper Upper Lower Lower 1% 5% 5% 1% 1% 5% 5% 1%

200 3-98 3-57 2-51 2-37 1000 3-41 3-26 2-76 2-68 250 3-87 3-52 2-55 2-42 1200 3-37 3-24 2-78 2-71 300 3-79 3-47 2-59 2-46 1400 3-34 3-22 2-80 2-72 350 3-72 3-44 2-62 2-50 1600 3-32 3-21 2-81 2-74 400 3-67 3-41 2-64 2-52 1800 3-30 3-20 2-82 2-76 450 3-63 3-39 2-66 2-55 2000 3-28 3-18 2-83 2-77 500 3-60 3-37 2-67 2-57 550 3-57 3-35 2-69 2-58 2500 3-25 3-16 2-85 2-79 600 3-54 3-34 2-70 2-60 3000 3-22 3-15 2-86 2-81 650 3-52 3-33 2-71 2-61 3500 3-21 3-14 2-87 2-82 700 3-50 3-31 2-72 2-62 4000 3-19 3-13 2-88 2-83 750 3-48 3-30 2-73 2-64 4500 3-18 3-12 2-88 2-84 800 3-46 3-29 2-74 2-65 5000 317 3-12 2-89 2-85 850 3-45 3-28 2-74 2-66 900 3-43 3-28 2-75 2-66 950 3-42 3-27 2-76 2-67 1000 3-41 3-26 2-76 2-68

31
APPENDIX B

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84.000000 5.0000000 .59519999 -.028643906

.63734391 4.0000000 .60699999 -.028743908

.63734391 11.000000 .60860000 -.023943909

.62915000 7.0000000 .60870000 -.026543908

.66844999 12.000000 .61040000 -.023543909

.62885945 10.000000 .61080000 -.024843909

.62885945 9:0000000 .61099999 -.025143906

.032401807 8.0000000 .61220000 -.026343912

.0035353269 6.0000000 .61250000 -.026943907

.14670000 80.000000 .61340000 .083956093

.021074478 78.000000 .61380000 .070456088 .0010498771 77.000000 .61409999 .064756088 5.0838811 76.000000 .61530000 .063056089

.63031229 75.000000 .61619999 .060756087

.64437553 81.000000 .61680000 .086856090

.028026464 82.000000 .61780000 .089456089

.038586470 84.000000 .61880000 .10445609

*-2. 4760868-04 83.000000 .61910000 .10335609

* 1.4412009-04 79.000000 .61920000 .082556091

-1.7180720 27.500000 .61970000 -.014843911

.089557514 35.000000 .62040000 -.011943907

47.000000 33.000000 .62069999 -.012143910

55.666667 38.000000 .62159999 -.010643907 3.8224483 25.000000 .62180000 -.015543908

* 3.6380990-04 17.000000 .62220000 -.019543909

.34652618 24.000000 .62249999 -.015743911

22.000000 20.000000 .62249999 -.018143907

62.000000 19.000000 .62310000 -.018243909 14.000000 31.000000 .62329999 -.012343906

33.476190 18.000000 .62500000 -.018543907

3.5094137 30.000000 .62520000 -.014043912

-5.5496990 27.500000 .62520000 -.014843911

.59509999 22.000000 .62520000 -.016943909

.74180000 23.000000 .62540000 -.016643912

3.7310967 16.000000 .62599999 -.020543911

5.9296834 13.000000 .62620000 -.023243912 53.536889 42.000000 .62670000 -.0082439110 34.208556 29.000000 .62760000 -.014243908

.087139798 26.000000 .62840000 -.015143909

* 1.8027863+02 33.000000 .62899999 -.012143910

53.536889 49.000000 .62909999 -.0065439120

.63734391 63.000000 .62920000 * 2.5609136-04

0. 52.000000 .62980000 -.0043439120

0. 47.000000 .63000000 -.0070439130

0. 46.000000 .63010000 -.0072439089

0. 65.000000 .63029999 .0016560927

0. 70.000000 .63070000 .0049560890

0. 45.000000 .63079999 -.0073439106

0. 36.000000 .63140000 -.011343911

36
COLUMN 10 COLUMN 11 COLUMN 12 COLUMN 13

5.0000000 43.000000 .63259999 -.0081439093

25.000000 44.000000 .63299999 -.0075439066

35.000000 41.000000 .63370000 -.0083439127

8.0000000 37.000000 .63380000 -.011143908

1.0000000 2.0000000 .63400000 -.042143911

0. 1.0000000 .63440000 -.042243913

0. 50.000000 .63440000 -.0059439093

4.0000000 72.000000 .63490000 .0066560879 4.0000000 71.000000 .63540000 .0065560937 2.0000000 51.000000 .63550000 -.0047439113

66.000000 .63609999 .0018560886 69.000000 .63730000 .0043560937 68.000000 .63760000 .0038560927 73.000000 .63829999 .015656091 67.000000 .63900000 .0037560910 60.000000 .63919999 -.0018439069

56.500000 .64110000 -.0029439107

74.000000 .64120000 .024956092 39.000000 .64170000 -.0097439066

48.000000 .64229999 -.0066439062

59.000000 .64390000 -.0019439086 21.000000 .64399999 -.017643906 14.000000 .65300000 -.022043906 55.000000 .66230000 -.0033439100 54.000000 .69809999 -.0035439059

40.000000 .70039999 -.0089439079

15.000000 .70209999 -.021143913

33.000000 .70779999 -.012143910

58.000000 .71990000 -.0024439096

56.500000 .72130000 -.0029439107

61.000000 .72420000 -.0012439117

62.000000 .72679999 *-4. 3906271-05

53.000000 .74070000 -.0036439076

64.000000 .74180000 * 9.5608830-04

37
LIST OF COMMANDS, DATA AND DIAGNOSTICS

DIMENSIOrvf 250 20 READ 1,2 3826 .6070 .09696 3836 .6087 .08583 3846 .6086 .1078 3856 .6134 .01329 3866 .6108 -.04528

3876 .6138 .1262 3886 .6125 -.04922

3896 .6122 -.07450

3906 .6110 .02926 3916 .6104 -.01599

3926 .7213 .02208 3936 .7078 .07162 3946 .7021 .04266

3956 .7004 .04604 3966 .6981 .03404 3976 .7242 .01416 3986 .7268 .03680 3996 .7418 .02983 4006 .7407 .01029 4016 .7199 .01746 4066 .6225 .03826 4076 .6254 .04812 4086 .6252 .01361 4096 .6267 -.02906

4106 .6218 .003828 4116 .6178 .04067 4126 .6216 .02104 4136 .6192 .003814 4146 .6191 -.1778 4246 .6250 -.04999 4256 .6188 .08734 4266 .6233 .01487 4276 .6225 .02140 4286 .6204 -.02172

4296 .6207 -.06341

4306 .6168 .01848 4316 .6141 -.07344

4586 .6291 -.02217

4596 .6231 .05215 4606 .6222 .1344 4616 .6252 .02088 4626 .6308 -.003182 4636 .6376 .04432 4646 .6330 -.03469 4656 .6303 -.06815

4666 .6301 -.02484 4676 .6390 .001270 4686 .6423 .007393

38
LIST OF COMMANDS, DATA AND DIAGNOSTICS

4696 .6300 -.002256

4706 .6260 .01151 4716 .6292 .05266 4726 .6298 .6804 4736 .6290 -.02382 4746 .6262 -.07195 4756 .5952 .08198 4766 .5951 .08620 4846 .6314 -.04368 4856 .6440 .03386 4866 .6439 .02495 4936 .6326 -.03362

4946 .6392 -.03325 4956 .6417 .01094 4966 .6412 -.04209

5036 .6530 -.004807

5046 .6411 .02106 5056 .6355 -.005957

5066 .6344 -.01485

5076 .6623 .09334 5236 .6276 .03809 5246 .6307 .01394 5256 .6354 .03019 5266 .6197 .03389 5276 .6153 .05951 5326 .6340 .01945

5316 .6338 .05811 5336 .6284 .07580 5366 .6162 -.05455

5376 .6252 -.04439

5396 .6349 .07879 5406 .6344 .07899

5416 .6361 .05468

5446 .6373 .05507 5456 .6337 .04438 5466 .6383 .01897 STATISTICAL 2,10,11,12,13 PRINT 10,11,12,13

STOP

NATIONAL BUREAU OF STANDARDS. WASH, D.C. 20234, OMNITAB II VERS 5.01 MAY 29.1971

39
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEETS

1. PUBLICATION OR REPORT NO. NBS TN 756

2. Gov't Accession No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE A Users' Guide to the OMNITAB Command "STATISTICAL ANALYSIS"

5. Publication Date March 1973

6. Performing Organization Code

7. AUTHOR(S)

H. H. Ku

8. Performing Organization

10. Project/Task/ Work Unit Nc 2050131

9. PERFORMING ORGANIZATION NAME AND ADDRESS NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

S ame as 9

13. Type of Report & Period CoveredFinal

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.)

This Technical Note is the first of a series of interpretive notes for a number of commands in the OMNITAB system that have the automatic printout feature.

Others planned in the series include commands FIT and POLYFIT, TWOWAY analysis,

ONEWAY analysis, and CORRELATION.

These notes aim to be self-contained so that users may have sufficient information on hand for the understanding of the statistics computed and to use them

for their immediate applications. Computation formulas are given in the text, and

a number of statistical tables are reproduced in the Appendix for the convenience

of users.

17. KEY WORDS (Alphabetical order, separated by semicolons) Computing, Statistical; documentation for

users; OMNITAB; statistics.

18. AVAILABILITY STATEMENT

(IT] UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

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