United States

Department of

Agriculture

Forest Service

Pacific Northwest

Research Station

General Technical

Report

PNW-GTR-456

March 1999

Sampling and

Estimating

Recreational Use

Timothy G. Gregoire and Gregory J. Buhyoff
Authors TIMOTHY G. GREGOIRE is with the School of Forestry and Environmental Studies,

Yale University, 360 Prospect Street, New Haven, Connecticut 06511; and GREGORY

J. BUHYOFF is with the Department of Forestry (0324), College of Forestry and Wildlife Resources, Virginia Polytechnic Institute and State University, Blacksburg, Virginia

24061. This work was performed under PNW P.O. 43-04H1-7-0159.
Gregoire, Timothy G.; Buhyoff, Gregory J. 1999. Sampling and estimating

recreational use. PNW-GTR-456. Portland, OR: U.S. Department of Agriculture,

Forest Service, Pacific Northwest Research Station. 39 p.

Probability sampling methods applicable to estimate recreational use are presented.

Both single- and multiple-access recreation sites are considered. One- and two-stage

sampling methods are presented. Estimation of recreational use is presented in a

series of examples.

Keywords: Sampling strategies, confidence intervals, stratified sampling.

This report is intended to provide guidance on the execution of sampling strategies to

estimate recreational use in settings such as National Forests. The intended audience

for the material presented here is resource managers, supervisors, and others in

positions of responsibility for the allocation and maintenance of recreational resources.

Informed management requires a credible assessment of how frequently and intensely

the recreation resource is used. The procedures described in this report are tools for

that purpose. The presentation is intended to be specific enough in the essential

characteristics that these sampling strategies can be applied generally to the tremendous diversity of recreation sites and forms of recreation supported by them. No

attempt has been made to anticipate special features of some recreation sites or

attributes of certain forms of recreation that would presume some modification of the

basic designs. As a consequence, this report serves more as a primer than as a

comprehensive guide.

The orientation of this report is quite general, too, in that we do not focus on particular

types of recreational-use areas or on types of recreational use, except for sake of

example. Because of the differing opportunities for sampling that are presented, we

distinguish broadly between overnight recreational-use areas and others. For lack of a

better or established terminology, we refer to the latter as “transient-recreational-use

areas.” For transient-recreational-use areas, we present sampling strategies for areas

with a single access or entry point separately from those suitable for areas with multiple

access or entry points.

Our concern is with estimating recreational use, as measured, for example, by the

number of weekday recreationists on a particular hiking trail. We have not attempted to

devise strategies to acquire information that cannot be observed by the person conducting the sample; that is, the strategies we propose are nonintrusive by design and

therefore insufficient to estimate, for example, the average age of recreationist on a

trail or to determine whether two hikers are members of a group hiking together. For

such purposes, the sample selection methods we present may remain applicable, but

the methods to elicit information would not.

Abstract

Summary
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Document continues on next page.
1

This report summarizes sampling strategies to estimate recreational use, such as

average number of hikers on a park or forest trail. The sampling and estimation plans

described in this report are aimed at producing credible estimates of visitor use based

on observed activity. Sampling to estimate duration of recreational use is a more

involved task than that considered here.

It would be futile to enumerate all possible forms of recreation amenable to the sampling strategies presented in this report, but hiking and overnight camping are two

forms that provide useful foci for discussion. Within a 24-hour-day sampling frame,

hiking along a specific trail can be regarded as a transient form of recreation, whereas

overnight camping is not. We broadly categorize sites as being a transient-use type or

an overnight-recreational-use type. Accordingly, sampling strategies for the two

categories of recreation sites are presented separately.

For recreation sites with a single access point (for example, a closed loop trail with a

lone trail head), the sampling ought to be conducted at that point. The neatness of the

sampling and estimation procedures deteriorates for sites having multiple access

points. We therefore treat the two types of sites separately, and we account for both

the sampling and estimation details applicable to each.

Each sampling plan has a random component, essential to guard against biased

estimation of recreational use. In other words, the sampling is probabilistic and therefore ensures that each recreationist using the site has an objective chance of being

included in the sample. This feature, lacking in self-selected samples of names written

in visitor registries, is crucial to the statistical validity of the results. Once a day and

time are selected for sampling a recreation site, it is imperative that sampling be

conducted (barring physical harm to the sampler, which may result from flash floods,

lightning strikes, or similar harsh weather and environmental conditions). Convenience

sampling must be avoided absolutely; for example, it is unacceptable to avoid sampling

on a randomly selected but rainy day.

Measurement error is an abiding concern in any sampling effort. The methods proposed in this report require that a trained observer record recreational use. Measurement error in this context is an incorrect count of the number of recreationists or a

misrecorded count of same. While more expensive, perhaps, than relying on mechanical or optical counters of visitor activity, the employment of a trained observer is more

reliable and informative. The control of measurement error within the probability

sampling framework imparts a scientific credibility to the system that will bear up under

close scrutiny.

The probabilistic sampling plan coupled with the proposed estimation procedures ensures

that recreational use will be unbiasedly estimated. The variance of the resulting estimate

is largely a function of sampling intensity and frequency. Past sample information may

provide data that can be used to plan the size of future samples at similar sites. This

topic is addressed further in “Planning a Sample of Recreational Use.”

The stratified random sampling methods suggested in this report are straightforward

to administer and can be applied virtually without modification to any recreation area.

We suggest, for example, that sampling be planned separately for weekends versus

weekdays, whenever recreational use varies much between these two weekly strata.

Introduction
2

If recreation or management activities occur much more, or less, frequently on holidays, then the resource manager (RM) might consider creating a third stratum, and

plan the sampling effort for that stratum separately from the other two.

Stratification is suggested as a way not only to obtain separate estimates for each

meaningfully defined stratum but also to increase the precision of the resulting estimates. Although more sophisticated plans can be devised and hold the promise of

estimating recreational use more precisely, they are not suggested because they lack

generality, are less intuitively plausible to persons untrained in the nuances of statistical

sampling, and are less easily transported and implemented elsewhere. For multipurpose surveys, simplicity of sample design is a virtue.

We focused principally on estimating the average number of persons engaging daily in

specific forms of recreation (hiking, biking, camping), during some period stipulated by

the RM. This period, which will comprise the sampling frame of sample days, can be a

month, a year, or longer. We refer hereafter to this designated period as a “season.”

Presumably, the season will be an interval having relevance to the management of the

recreational resource. However it is defined, we explain how to conduct the sample

and show how to prorate the sample data to obtain an estimate of recreational use

applicable to that period, either daily or seasonally.

We have woven several numerical examples into the text. Their purpose is to take

some of the mystery out of the symbolic formulas for estimators of recreational use by

demonstrating how the sample data actually are manipulated. The data are fictitious,

as the examples needed to be short to encourage verification by those using the

procedures. A simple computer spreadsheet should be sufficient to process not only

the data provided in the examples but actual sample data as well.

We restricted the sample selection method to simple random sampling without replacement of days. In this section, we describe two methods that use a computer spreadsheet to select a simple random sample without replacement (SRS). Alternative

methods of selecting a SRS may, of course, be used instead.

Suppose that the RM wishes to estimate recreational use for weekends separately from

weekdays. The technique of poststratification can be used to do this when stratification

has not been done prior to sampling. Poststratification procedures are exceedingly

valuable in “after the fact” analyses of sample data. The treatment of poststratification

is beyond the scope of this report, however. We suppose instead that the RM has the

liberty to select sample days, by strata, beforehand. In the following, we first outline a

procedure to select stratified random samples for the particular case where the RM is

interested in estimating annual recreation (population of 365 or 366 days). Then we

broaden this to the case where the time period of interest is defined differently, such

that the population consists of some number of days implied by the definition, which we

show as N.

Suppose that the RM had been interested in sampling during the 1997 calendar year, in

which there were 111 weekend days, including holidays, and 254 weekdays. Based on

considerations shown in the appendix, we assumed that the RM decided to sample 6

percent of the weekend and 3 percent of the weekdays throughout the year. After

rounding to the nearest integer value, the RM needs to randomly select (in the sense of

Selecting Sample

Days
3

selecting with equal probability) 0.06 × 111 = 7 weekend days and 0.03 × 254 = 8

weekdays. In a spreadsheet, generate a column sequence of numbers from 1 to 365.

Most spreadsheets have the facility to generate such a sequence automatically, and

the chore of entering 365 numbers can be avoided. In an adjacent column, generate a

series of uniform random numbers from the spreadsheet’s function list.

With the spreadsheet’s sorting function, rearrange both columns in order of increasing

value of the random numbers in the second column.

From the sorted list, select from the top of the first column the first 8 days that correspond to weekdays. Then select the first 7 days that are weekend days. You will have

chosen a SRS of 8 weekdays from the pool of 254 and a SRS of 7 weekend days from

the pool of 111.

Many spreadsheets have a SRS function that can be used instead. If so, it will be

necessary to list separately the weekdays from the weekend days, and to separately

invoke the SRS function on each list. We think the procedure described first is easier

but want to point out the option.

Once a day has been selected in advance, recreational use must be recorded on that

day. Sampling must not be postponed owing to inclement weather, unless the

sampler’s safety is endangered. In that case, another day for sampling must be

selected at random; in particular, it is not acceptable to choose the day immediately

following the initially selected day. An advantage of the selection procedure using the

sorted list of random numbers is that replacement sample days can be chosen easily

by going further down the sorted list.

Now consider the case when the period of interest is the season spanning Memorial

Day to Thanksgiving Day. Measured in days, the length of this period was N = 180

days altogether, in calendar year 1997; of these, there were N1

 = 126 weekdays and

N2 = 54 weekend days and holidays. We suppose that α1 = 0.03 remains the proportion

of weekdays to be sampled; that is, the percentage of weekdays to be selected is 100%

× α1 = 3%, as before. Similarly, α2 = 0.06 remains the proportion of weekend days to be

sampled, so that the percentage of weekend days to be selected is 100% × α2 = 6%.

Regardless of the numbers attached to the symbols, the general formula for computing

the actual numbers of weekdays to be selected is,

n1 = α1N1 ,

where α1 represents the sampling proportion from the stratum and N1 represents the

number of days in stratum 1. Similarly for stratum 2, the general formula for computing

the actual numbers of weekend days to be selected is,

n2 = α2N2 ,

where α2 represents the sampling proportion from this stratum and N2 represents

the number of days in stratum 2 . Using the hypothetical numbers put forth for this

example, these formulas solve to
4

n1 = α1N1 = 0.03 × 126 = 4 weekdays

and

n2 = α2N2 = 0.06 × 54 = 3 weekend days ,

after rounding the answers to the closest integer value.

To select the actual days to conduct the sampling, use the same spreadsheet algorithm

outlined above when the entire calendar year was the sampling frame. Instead of N =

365 as in that case, here the column sequence needs to extend from 1 to N = 180,

where day 1 implicitly represents Memorial Day. More generally, day 1 represents the

initial day of the period of interest.

When holidays are partitioned separately into a third stratum, as mentioned in the

“Introduction,” the formulas presented above extend rather naturally; namely, the RM

must determine the number of holidays, N3, comprising this third stratum, as well as the

sampling fraction, α3, to be included in the sample. The number of days selected from

stratum 3 is

n3 = α3N3 .

The above discussion is framed in terms of a sampling day, yet it is unrealistic to

assume that this coincides with a 24-hour day when recreational use of the site takes

place. The RM must decide, for each particular site, the period during the 24-hour day

when recreational use of the site takes place. This will differ, perhaps, with the form of

recreational use being monitored. In some cases the 14-hour period from 6:00 a.m. to

8:00 p.m. may be sufficient; in other cases, it may be excessive or insufficient. In

principle, anything less than a 24-hour sampling day invites measurement (observer)

error. We assume that the RM chooses the sampling day knowledgeably to minimize

the magnitude of measurement error.

As an aside, it might be worthwhile to note that the cost of sampling can be computed

rather quickly in the case where sampling costs do not differ among strata: it costs as

much to station a sampler during the week as it does during the weekend or on a

holiday. In this case, the total size of the sample is n = n1

 + n2 or n = n1+ n2 + n3, depending on the number of strata one decides to use. If c represents the cost of sampling for

a single day, then the overall cost of sampling is just

C = c × n .

For sake of discussion, we presume that there are two types of recreational use one

wishes to monitor—backpacking and biking—separately across two strata—weekdays

and weekend days. Our use of two different forms of recreational use does not imply

more or fewer forms cannot be monitored. Our use of two is solely for example.

On randomly selected days, a person will observe and record use of the site. This

person should be located at the access point to the site. Ideally, this person should

Sampling Strategy

for TransientRecreational-Use

Areas

Single Access Sites
5

record recreational use for the entire portion of the day during which recreational

activities are likely to occur. If sampling cannot be conducted for an entire day, then a

two-stage sampling approach should be adopted, wherein days are selected randomly

as described above, and then a predetermined block of hours within the day is selected

randomly. Details of the two-stage approach are discussed in “Two-Stage Selection,”

on page 14.

The observer stationed at the access point should record only traffic entering the site.

The data recording form, whether paper or electronic, should have fields to indicate the

recreation site, sampling location within site, date, and the beginning and ending times

of observation.

Each person entering the trail should be recorded. At a minimum, separate tallies by

recreational use must be maintained, in this case backpacking and biking. Further

information may prove useful for management purposes; for example, tallying

recreationists by both use and gender, providing gender can be discerned with negligible measurement error.

As established in “Selecting Sample Days,” N, N1

, and N2 represent, respectively, the

number of days in the period of interest, in the subset of the period of interest encompassing weekdays, and in the subset encompassing weekend days. Similarly, n, n1

,

and n2 represent the number of sample days.

To distinguish among the number of individual backpackers tallied on the different

sample days in stratum 1, we introduce this notation:

I1,1(backpackers) = observed number of backpackers on sample day 1 in stratum 1

I1,k(backpackers) = observed number of backpackers on sample day k in stratum 1

I1,n1(backpackers) = observed number of backpackers on sample day n1 in stratum 1

Similarly for the tallies of backpackers in stratum 2, we have the following:

I

2,1(backpackers) = observed number of backpackers on sample day 1 in stratum 2

I2,k(backpackers) = observed number of backpackers on sample day k in stratum 2

I2,n2(backpackers) = observed number of backpackers on sample day n2 in stratum 2

The numbers of bikers tallied throughout the sample can be symbolized in a parallel

fashion:

I1,1(bikers) = observed number of bikers on sample day 1 in stratum 1

I1,k(bikers) = observed number of bikers on sample day k in stratum 1

I1,n1(bikers) = observed number of bikers on sample day n1 in stratum 1

Notation
6

For stratum 2,

I

2,1(bikers) = observed number of bikers on sample day 1 in stratum 2

I

2,k(bikers) = observed number of bikers on sample day k in stratum 2

I

2,n2

(bikers) = observed number of bikers on sample day n2 in stratum 2

Estimating formulas by stratum—An estimate of the number of backpackers per day

in stratum 1 is given by,

so that an estimate of the total number of backpackers using this site in stratum 1 is

this average multiplied by the number of days in the stratum:

An estimate of the number of backpackers per day in stratum 2 is given by,

The estimate of total number of backpackers in stratum 2 is,

I ˆ 

1 ( backpackers) = 1 

n 1 

n 

1 

3 

k = 1 

I 1 , k ( backpackers) 

= 

I 1 , l ( backpackers) + ˇ + I 1 , k ( backpackers) + ˇ + I 1 , n 1 

( backpackers) 

n 1 

 , 

T ˆ 

1 ( backpackers) = N1 H I ˆ 1 ( backpackers) 

I ˆ 

2 ( backpackers) = 1 

n 2 

n 2 

3 

k = 1 

I 2 , k ( backpackers) 

= 

I 2 , l ( backpackers) + ˇ + I 2 , k ( backpackers) + ˇ + I 2 , n 2 

( backpackers) 

n 2 

 . 

T ˆ 2 ( backpackers) = N2 H I ˆ 

2 ( backpackers).
7

An estimate of the average number of bikers per day in stratum 1 is given by,

so that the estimate of the total number of bikers using this site in stratum 1 is this

average multiplied by the number of days in the stratum,

An estimate of the average number of bikers per day in stratum 2 is given by,

The estimate of total number of bikers in stratum 2 is,

Estimating formulas for both strata combined—Because the estimated total number

of backpackers in both strata is the sum,

the average number of backpackers per day in both strata is estimated as

I ˆ 

1 ( bikers) = 1 

n 1 

n 

1 

3 

k = 1 

I 1 , k ( bikers) 

= 

I 1 , l ( bikers) + ˇ + I 1 , k ( bikers) + ˇ + I 1 , n 1 

( bikers) 

n 1 

 , 

T ˆ 

1 ( bikers) = N1 H I ˆ 1 ( bikers) 

I ˆ 

2 ( bikers) = 1 

n 2 

n 2 

3 

k = 1 

I 2 , k ( bikers) 

= 

I 2 , l ( bikers) + ˇ + I 2 , k ( bikers) + ˇ + I 2 , n 2 

( bikers) 

n 2 

 . 

T ˆ 2 ( backpackers) = N2 H I ˆ 

2 ( bikers).

T ˆ ( backpackers) = T ˆ 

1 ( backpackers) + T ˆ 2 ( backpackers) , 

I ˆ ( backpackers) = T ˆ ( backpackers) / N .
8

For bikers a similar computation is used:

The average number of bikers per day in both strata is estimated as

Estimating the total number of recreationists per season is straightforward:

The average number of both types of recreationists per day is estimated by

Example 1—Management needed to estimate the amount of backpacking and biking

recreation on the Spruce Run Trail throughout the 1997 season from Memorial Day to

Thanksgiving Day. During this period, sampling was conducted on n1 = 4 weekdays

and n2 = 3 weekend days. The observed use is summarized in the following tabulation:

Recreationist Stratum 1: weekdays Stratum 2: weekends

Backpackers 13 16 12 11 28 32 22

Bikers 8 6 8 10 10 18 14

From these daily tallies of recreational use, the average daily number of weekday

backpackers can be estimated by,

Similarly, the estimate of the average daily number of weekend day backpackers is,

T ˆ ( bikers) = T ˆ 

1 ( bikers) + T ˆ 2 ( bikers) . 

I ˆ ( bikers) = T ˆ ( bikers) / N .

I ˆ = T ˆ / N .

T ˆ = T ˆ ( backpackers) + T ˆ ( bikers).

I ˆ 

1 ( backpackers) = 

1 

4 á 13 + 16 + 12 + 11 é = 13 .

I ˆ 

2 ( backpackers) = 

1

3 á 28 + 32 + 22 é = 24 .
9

The estimates of biking use are computed as,

and

Because N = 180 days in this period in 1997, the average daily number of backpackers

is estimated to be,

where 126 and 54 represent, respectively, the number of days in the weekday and

weekend strata. Seasonally, the estimated number of backpackers is,

Likewise, the average daily number of bikers is,

Seasonally, the estimated number of bikers is,

The total number of both types of recreationists throughout the season is,

The average daily number of backpackers and bikers is

I ˆ 

1 ( bikers) = 

1 

4 á 8 + 6 + 8 + 10 é = 8 , 

I ˆ 

2 ( bikers) = 

1 

3 á 10 + 18 + 14 é = 14 . 

I ˆ ( backpackers) = 1 

180 á 126 H 13 + 54 H 24 é = 16. 3 backpackers , 

T ˆ ( backpackers) = 2934 .

T ˆ ( bikers) = 1764 .

T ˆ = 4698 .

I ˆ ( bikers) = 1 

180 á 126 H 8 + 54 H 14 é = 9 . 8 bikers . 

I ˆ = 26. 1 .
10

Stratum estimates—An estimate of the variance of Î(backpackers) depends on the

extent to which the individual daily tallies, I

1,1, I1,2,..., I1,n1

 vary around Î(backpackers).

Indeed, it is computing the variance of an estimate of recreation visits and constructing

a confidence interval for the underlying population value of the number of visits that

necessitate keeping the daily tallies separate. As a computational aid, we suggest the

these deviations be computed separately as,

The estimated variance of Î is designated as vâr[Î1(backpackers)] and can be computed

by the following formula:

The estimated standard error of

Confidence intervals for average number of recreationists visiting a site—An

approximate 90-percent confidence interval for I1(backpackers) is constructed as,

where t

df

0.90 is the 90th percentile point from the Student’s t distribution with df = n1

 – 1

degrees of freedom.

An alternative expression of the confidence interval which many find more appealing is

one for which the

Variance Estimation and

Confidence Intervals

DIl , 1 ( backpackers) = I l , 1 ( backpackers) − I ˆ 

1 ( backpackers) 

!

DIl , k ( backpackers) = I l , k ( backpackers) − I ˆ 

1 ( backpackers) 

!

DIl , n 1 ( backpackers) = I l , n 1 ( backpackers) − I ˆ 

1 ( backpackers).

v a ˆr I ˆ 

l ( backpackers) = 

ä 

ã å å 1 

n 1 

− 1 

N 1 

ë 

í ì ì 

n 1 

3 

k = 1 

DI 2 

1 , k ( backpackers) 

n 1 − 1 .

I ˆ 

1 is v a ˆr I ˆ 1 ( backpackers) .

I ˆ 

1 ( backpackers) " t 0 . 9 0

df v a ˆr I ˆ 

1 ( backpackers) , 

" t 0 . 9 0

df v a ˆr I ˆ 

1 ( backpackers) 
11

portion is expressed as a percentage of Î1(backpackers):

where

For the weekend stratum,

where DI2,k(backpackers) = I2,k(backpackers) - Î2

(backpackers), and so on.

An approximate 90-percent confidence interval for I2

(backpackers) is constructed as,

where t

df

0.90 is the 90th percentile point from the Student’s t distribution with df = n2

 – 1

degrees of freedom. Equivalently, one can express the confidence interval as

Î2(backpackers) ± E% where

Combined strata estimates—For both strata combined,

An approximate 90-percent confidence interval for I(backpackers) is constructed as,

I ˆ 

1 ( backpackers) " E% ,

E % = 

:

; 

<

= = =

= = = 

t 0 . 9 0

df v a ˆr I ˆ 

1 ( backpackers) 

I ˆ 

1 ( backpackers) 

B

C 

D

E E E

E E E 100% . 

v a ˆr I ˆ 

2 ( backpackers) = 

ä 

ã å å 1 

n 2 

− 1 

N 2 

ë 

í ì ì 

n 

2 

3 

k = 1 

DI 2 

2 , k ( backpackers) 

n 2 − 1 , 

I ˆ 

2 ( backpackers) " t 0 . 9 0

df v a ˆr I ˆ 

2 ( backpackers) 

E % = 

:

; 

<

= = =

= = = 

t 0 . 9 0

df v a ˆr I ˆ 

2 ( backpackers) 

I ˆ 

2 ( backpackers) 

B

C 

D

E E E

E E E 100% .

v a ˆr I ˆ ( backpackers) = 1 

N 2 7 N 2 

1 H va ˆr I ˆ 

1 ( backpackers) + N2 

2 H va ˆr I ˆ 

2 ( backpackers) ? .

I ˆ ( backpackers) " t 0 . 9 0

df v a ˆr I ˆ ( backpackers) 
12

where t

df

0.90 is the 90th percentile point from the Student’s t distribution with df = n1

 – n2 – 2

degrees of freedom. An equivalent expression is Î(backpackers) ± E%,

where

Example 2—Continuing from example 1, a few spreadsheet manipulations provided,

Approximate 90-percent confidence intervals are constructed with

Parameter Approximate 90-percent confidence interval

I

1

(backpackers) 13 ± 2.5 backpackers ≡ 13 backpackers ± 19%

I2(backpackers) 24 ± 8.2 backpackers ≡ 24 backpackers ± 34%

I(backpackers) 16.3 ± 2.3 backpackers ≡ 16.3 backpackers ± 14%

The corresponding calculations for bikers are:

Approximate 90-percent confidence intervals are constructed with

E % = 

: 

; 

< 

= = 

= = = 

t 0 . 9 0

df v a ˆr I ˆ ( backpackers) 

I ˆ ( backpackers) 

B 

C 

D 

E E 

E E E 100% .

v a ˆr I ˆ 

1 ( backpackers) = 

ä 

ã å å 1 

4 − 1 

126

ë 

í ì ì á 4 . 6  ̄ é = 1 . 13

v a ˆr I ˆ 

2 ( backpackers) = 

ä 

ã å å 1 

3 − 1 

54

ë 

í ì ì á 25. 3  ̄ é = 7 . 98

v a ˆr I ˆ ( backpackers) = 1 

1802 1262 

 H 1 . 13 + 542  H 7 . 98 = 1 . 27 .

v a ˆr I ˆ 

1 ( bikers) = 

ä 

ã å å 1 

4 − 1 

126

ë 

í ì ì á 2 . 6  ̄ é = 0 . 65

v a ˆr I ˆ 

2 ( bikers) = 

ä 

ã å å 1 

3 − 1 

54

ë 

í ì ì á 16. 0 é = 5 . 04

v a ˆr I ˆ ( bikers) = 1 

1802 1262 

 H 0 . 65 + 542  H 5 . 04 = 0 . 77

t 0 . 9 0

3 = 2 . 353, 

t 0 . 9 0

2 = 2 . 290, and t 0 . 9 05 = 2 . 015: 

t 0 . 9 0

3 = 2 . 353, 

t 0 . 9 0

2 = 2 . 290, and t 0 . 9 05 = 2 . 015: 
13

Parameter Approximate 90-percent confidence interval

I

1

(bikers) 8 ± 1.9 bikers ≡ 8 bikers ± 34%

I2(bikers) 14 ± 6.5 bikers ≡ 14 bikers ± 47%

I(bikers) 9.9 ± 1.8 bikers ≡ 9.9 bikers ± 18%

Confidence intervals for totals—Because

then

and an approximate 90-percent confidence interval for T1(backpackers) is just

where df = n1 – 1.

Similarly,

and

when df = n2 – 1.

Finally,

and

T ˆ 

1 ( backpackers) = N1 H I ˆ 1 ( backpackers) , 

v a ˆr T ˆ 

1 ( backpackers) = N2 

1 H v a ˆr I ˆ 

1 ( backpackers) , 

v a ˆr T ˆ 2 ( backpackers) = N2 

2 H v a ˆr I ˆ 

2 ( backpackers) 

T ˆ 

2 ( backpackers) " N2 H t 0 . 9 0

df v a ˆr I ˆ 

2 ( backpackers) , 

v a ˆr T ˆ ( backpackers) = va ˆr T ˆ 

1 ( backpackers) + v a ˆr T ˆ 2 ( backpackers) 

T ˆ 

1 ( backpackers) " N1 H t 0 . 9 0

df v a ˆr I ˆ 

1 ( backpackers) , 
14

where df = n1 + n2 – 2.

A parallel development holds when estimating the total number of recreational bikers,

namely T(bikers).

Example 3—Using the results from example 2, the following intervals for T1(backpackers),

T2(backpackers), and T(backpackers) are obtained from the above formulas:

Parameter Approximate 90-percent confidence interval

T1(backpackers) 1638 ± 315 backpackers ≡ 1638 backpackers ± 19%

T2

(backpackers) 1296 ± 445 backpackers ≡ 1296 backpackers ± 34%

T(backpackers) 2934 ± 409 backpackers ≡ 2934 backpackers ± 14%

Notice that the ± (percentage) terms are identical to those in confidence intervals for

I1(backpackers), I

2

(backpackers), and I(backpackers).

If the staffing and expense of sampling recreational use for an entire day are prohibitive, we suggest a two-stage approach, wherein each potential day of sampling is

partitioned into fixed blocks of time. For a 14-hour sampling day, four 3.5-hour blocks

might make a logical partition. Three 5-hour blocks, likewise, would make sense for a

15-hour sampling day. The two-stage selection procedure consists of the random

selection of a day, followed by a random selection of a block of time within that day.

We envision that the number of blocks into which each day is partitioned will be rather

small; for example, two to four blocks per day. Sampling of recreational use on the trail

then proceeds as previously explained.

The estimation formulas presented earlier are appropriate only for single-stage sampling wherein the entire day is selected. Rather than repeat so many formulas here, we

describe a multiplicative adjustment to the previous formulas. Suppose that the

sampling day is partitioned such that each block of the partition represents a fraction, f,

of the entire day, and that a single block of hours within a sampling day constitutes the

subsample. Then for both backpackers and bikers, the estimation formulas for Î1 and Î2

should be multiplied by the factor 1/f. The estimation formulas for Î, Tˆ

1

, Tˆ

2

, and Tˆ

remain unchanged.

Unless more than one block within a day is subsampled, there is no way to unbiasedly

estimate the variance of Î1

, Î2, Î, Tˆ

1

, Tˆ

2

, and T.ˆ The variance estimators given in the

preceding section can be used, but they will underestimate the actual amount of

sampling variation. An obvious alternative is to subsample at least two of the blocks of

time into which the sample day is partitioned. In view of the coarse partitioning of each

sampling day that we envision, however, the rationale for subsampling at least two

blocks within a day, rather than sampling the entire day, becomes rather strained.

If the two-stage approach is adopted, it is essential that each random selection of a

sampling day be accompanied by a random selection of a time block within that day. It

is not appropriate to randomly select a single time block, which then is used throughout

all days of sampling.

Two-Stage Selection

T ˆ ( backpackers) " N H t 0 . 9 0

df v a ˆr I ˆ ( backpackers) , 
15

It is important that the two-stage procedure, if adopted, be used throughout the entire

sampling effort; moreover, the partitioning of the day into blocks of consecutive hours

must remain constant. We envision a partition of each day, for example, into four

blocks where the first block of time spans 6:00-9:30 a.m.; the second spans 9:30 a.m.

to 1:00 p.m.; the third spans 1:00-4:30 p.m.; and the fourth spans 4:30-8:00 p.m.

Under the two-stage selection, this partitioning of the day is fixed, and one of these four

time blocks would be chosen at random. It is not acceptable, for example, to randomly choose any time of day as the start of a 3.5-hour block of time. It is crucial that

sampling be conducted during the selected block of time. Subjective substitution of

one time block for another (e.g., due to inclement weather) will inject bias into the

estimates of recreational use.

It is not important that a two-stage procedure be used for all trails, but it would simplify

record keeping if the same sample design were implemented consistently for all trails.

When recreationists can enter and exit a site at more than one access point, it is

possible to unbiasedly estimate recreational use by stationing an observer at each

point on the selected sampling days. Each observer would record the number of

recreationists entering the site, presuming again that separate tallies would be maintained for backpackers and bikers. In this situation, recreationists passing by the

access point would not be tallied, as they would have been tallied at the entry. Recreation use may be estimated by combining all the observers’ tallies into a single tally,

and then using the formulas given for “single-access sites.” Very likely, this option is

not feasible for sites having numerous access points.

A two-stage sample is an alternative when there are fewer observers available than

there are access points. One consideration would be to randomly (that is, simple

random sampling without replacement) assign the available observers among the

access points in a two-stage sampling scheme. Suppose m represents the number of

access points at which observers are stationed on a particular sampling day, and that

not only is m identical for all strata but its value also remains constant throughout the

sampling season. It is understood, however, that the set of selected m access points

will differ from one sampling day to another.

Under this two-stage setup, suppose that

M = the number of access points on the site

so that

g = m/M.

Furthermore let

I

1,1,1(backpackers) = observed number of backpackers on sample day 1 in stratum 1 at

the first of the subsampled access points

Multiple Access Sites
16

I1,1,j (backpackers) = observed number of backpackers on sample day 1 in stratum 1 at

the jth of the subsampled access points

I

1,1,m (backpackers) = observed number of backpackers on sample day 1 in stratum

1 at the mth of the subsampled access points

Similarly for the tallies of backpackers in stratum 2, we have

I

2,1,1 (backpackers) = observed number of backpackers on sample day 1 in stratum 2 at

the first of the subsampled access points

I2,1,j (backpackers) = observed number of backpackers on sample day 1 in stratum 2 at

the jth of the subsampled access points

I2,1,m (backpackers) = observed number of backpackers on sample day 1 in stratum 2 at

the mth of the subsampled access points

The tallies for bikers for each day, stratum, and access point can be denoted in the

same way.

Let Î1,k(backpackers) represent the estimated average number of individual backpackers

for the kth sampling day in stratum 1, which is computed as,

Similarly for stratum 2,

As for the single-access point section, let I1

 and I2 represent the average daily recreational use of the trail for strata 1 and 2, respectively. The two-stage estimation

formulas for I1 and I2 are,

and

I ˆ 

l , k ( backpackers) = 

M 

m 

m 

3 

j = 1 

I l , k , j ( backpackers).

I ˆ 

2 , k ( backpackers) = 

M 

m 

m 

3 

j = 1 

I 2 , k , j ( backpackers).

I ˆ 

1 ( backpackers) = 1 

n 1 

n 

1 

3 

k = 1 

I ˆ 

1 , k ( backpackers) 

I ˆ 

2 ( backpackers) = 1 

n 2 

n 2 

3 

k = 1 

I ˆ 

2 , k ( backpackers).
17

Formulas for Tˆ

1

(backpackers), Tˆ

1

(backpackers), Î(backpackers), and Tˆ(backpackers),

remain unaltered from those given in the preceding section.

The variance of Î

l

(backpackers) comes from two sources: the first-stage component of

variance that results from observing recreational use on only a fraction, n1 of N, of all

the days of the season; and the second-stage component of variance that results from

observing recreational use at only a fraction, g, of all of the M access points for the site

on any given sampling day.

Just as when working up the variance estimators for the single-stage sampling, it

proves convenient to compute deviations from the average. In particular, suppose the

deviation of Î

l,k,j(backpackers) from the estimated average number of backpackers using

the site on the kth day of sampling is expressed as,

Because there are m access points in the subsample and n1 sample days in the first

stage of sampling, there are m × n1 deviations of the form DIl,k,j.

Additionally, suppose the deviation of Î

l,k(backpackers) from the estimated daily average recreational use per access point is expressed as,

DÎl,k = Îl,k(backpackers) – Îl,(backpackers) .

Because there are n1 sample days, there are n1 deviations of the form DIl,k.

Similarly the deviations in stratum 2 are designated as,

and

DI2,k = Î2,k(backpackers) – Î2(backpackers) .

There are m × n2 deviations of the form DI2,k,j and there are n2 deviations of the form DÎ2,k.

With these, the variance of Î(backpackers) is estimated as,

+ 

n 1 M 2 

N 1 

ä 

ã å å 1 

m − 1 

M 

ë 

í ì ì 

n 1 

3 

k = 1 

m 

3 

j = 1 

DI 2 

1 , k , j ( backpackers) 

n 1 á m − 1 é .

DIl , k , j = I l , k , j ( backpackers) − 1 

m 

m 

3 

i = 1 

I l , k , i .

DI 2 , k , j = I 2 , k , j ( backpackers) − 1 

m 

m 

3 

i = 1 

I 2 , k , i , 

ä 

ã å å 1 

n 1 

− 1 

N 1 

ë 

í ì ì 

n 1 

3 

k = 1 

D I ˆ 2 

1 , k ( backpackers) 

n 1 − 1 v a ˆr 2 s I ˆ 

1 ( backpackers) = 
18

Notice that the size of the first stage sample, n1 appears in the denominator of both

variance terms, whereas the second-stage sample size, m, appears only in the denominator of the second. This implies that the variance of a two-stage sample reduces

more quickly by enlarging the first stage sample than the second. In other words, if

there is no difference in the cost of either sampling on an additional day or sampling an

additional access point on a given day, sample on an additional day to increase the

size of the first-stage sample.

For stratum 2,

As before,

and

All the above can be developed in parallel fashion to estimate the amount of bikers

recreating on the site.

v a ˆr 2 s I ˆ 

2 ( backpackers) = 

ä 

ã å å 1 

n 2 

− 1 

N 2 

ë 

í ì ì 

n 

2 

3 

k = 1 

D I ˆ 2 

2 , k ( backpackers)

n 2 − 1 

+ 

n 2 M 2 

N 2 

ä 

ã å å 1 

m − 1 

M 

ë 

í ì ì 

n 2 

3 

k = 1 

m 

3 

j = 1 

DI 2 

2 , k , j ( backpackers) 

n 2 á m − 1 é 

T ˆ 

1 ( backpackers) = N1 H I ˆ 1 ( backpackers) 

v a ˆr T ˆ 

1 ( backpackers) = N2 

1 H v a ˆr I ˆ 

1 ( backpackers) 

T ˆ 

2 ( backpackers) = N2 H v a ˆr I ˆ 2 ( backpackers) 

v a ˆr T ˆ 

2 ( backpackers) = N2 

2 H I ˆ 

2 ( backpackers) 

T ˆ ( backpackers) = T ˆ 

1 ( backpackers) + T ˆ 2 ( backpackers) 

v a ˆr T ˆ ( backpackers) = va ˆr T ˆ 

1 ( backpackers) + v a ˆr T ˆ 2 ( backpackers) 

I ˆ ( backpackers) = T ˆ ( backpackers) / N 

v a ˆr I ˆ ( backpackers) = va ˆr T ˆ ( backpackers) / N 2 
19

Construction of confidence intervals for population parameters based on estimates

from two-stage and more complex designs is problematic. Since the introduction of

resampling techniques, such as the jackknife and the bootstrap, there has been a

tremendous amount of research into the construction of confidence intervals using

these tools. Any useful treatment of this topic is outside the scope of the present

report, however.

Example 4—In this example we expand on the previous three examples by supposing

that the data provided in example 1 were collected at one randomly selected access

point on each of the 7 days of sampling (4 weekdays and 3 weekend days). These

data are augmented with those collected on the same sampling days but from a second

randomly selected access point; that is, in this example, we demonstrate how to

estimate backpacker and biker activity on the Spruce Run Trail based on observations

from a two-stage sample of m = 2 access points. Purely for the sake of example, we

assume that the trail has a total of M = 10 access points from which m = 2 were selected

for each day of sampling. Hence, the second-stage sampling fraction is g = 0.2.

The sample data are presented in table 1. The first value in each cell was obtained

from one of the randomly selected access points on that sample day, and the other was

obtained from the second of the selected access point.

Table 1—Two-stage sample observations for each day and access point

Recreationist -------- Stratum 1: weekdays-------- -- Stratum 2: weekends --

Backpackers 13, 17 16, 17 12, 9 11, 21 28, 35 32, 32 22, 28

Bikers 8, 11 6, 7 8, 11 10, 8 10, 12 18, 17 14, 19

Based on these observations,

The remaining calculations of Î1,k and Î

2,k for backpackers and bikers are summarized in

table 2.

To compute estimates of recreational use by gender

(or any other sociodemographic breakdown that is

recorded), substitute the “number of individuals”

described on the preceding pages with the “number of

individuals of the gender of interest.”

I ˆ 

1 , l ( backpackers) = 

10

2 á 13 + 17 é = 150 .
20

Table 2—Daily averages of recreational

use within strata

Stratum Backpackers Bikers

Stratum 1:

Î1,1 150 95

Î1,2 165 65

Î1,3 105 95

Î1,4 160 90

Stratum 2:

Î2,1 315 110

Î2,2 320 175

Î2,3 250 165

These values, in turn, enable one to estimate the average daily recreational use by

stratum; for example,

The remaining trail estimates of recreational use, Î1

, and Î

2

, for backpackers and bikers

are summarized in table 3.

Table 3—Estimated daily recreational

use for entire traila

Stratum Backpackers Bikers

Stratum 1:

Î1 145.0 86.25

Stratum 2

Î2 295.0 150.0

aAverages per access point are obtained by

dividing these values by M =10.

Because Î

1,1(backpackers) = 15.0 and Î2,1(backpackers) = 31.5, the deviations of each

access point observation can be computed from the daily average as:

 DI1,1,1 = 13 – 15 = –2

 DI1,1,2 = 17 – 15 = 2

 DI2,1,1 = 28 – 31.5 = –3.5

 DI2,1,2 = 35 – 31.5 = 3.5

and so on, as summarized in table 4,

I ˆ 

1 ( backpackers) = 

1 

4 á 150 + 165 + 105 + 160 é = 145 .
21

Table 4—Deviations of each observation (shown in table 1) from daily average at

each access point

Stratum -------------Backpackers ----------- ----------------Bikers ---------------

Access point Access point Access point Access point

Stratum 1: j = 1 j = 2 j = 1 j = 2

DI1,1,j -2.0 2.0 -1.5 1.5

DI1,2,j -0.5 0.5 -0.5 0.5

DI1,3,j 1.5 -1.5 -1.5 1.5

DI1,4,j -5.0 5.0 1.0 -1.0

Stratum 2:

DI2,1,j -3.5 3.5 -1.0 1.0

DI2,2,j 0.0 0.0 0.5 -0.5

DI2,3,j 3.0 -3.0 -2.5 2.5

The sum of these squared deviations for each stratum and recreational use are shown

in table 5.

Table 5—Squared deviations of each observation from the daily averages,

summed within strata

Stratum Backpackers Bikers

Stratum 1 Sum of squared deviations, DI1,k,j Sum of squared deviations, DI1,k,j

63.0 11.5

Stratum 2 Sum of squared deviations, DI2,k,j Sum of squared deviations, DI2,k,j

42.5 15.0

The deviations denoted by DÎ1,k(backpackers) are the differences between each Î

1,k

value and Î

1

. For DÎ1,1(backpackers),

DÎ1,1(backpackers) = 150 – 145 = 5.0 ,

and for DÎ1,1(bikers),

DÎ1,1(bikers) = 95 – 86.25 = 8.75 .

The remaining deviations for backpackers and bikers are displayed in table 6.
22

Table 6—Deviations of daily estimates

from overall daily average within strata

Stratum Backpackers Bikers

Stratum 1:

DÎ1,1 5 8.75

DÎ1,2 20 -21.25

DÎ1,3 -40 8.75

DÎ1,4 15 3.75

Stratum 2:

DÎ2,1 20 -40.00

DÎ2,2 25 25.00

DÎ2,3 -45 15.00

The sum of squared deviations are needed in the variance formulas, along with the

values of n1, N1, n2, N2, m, M, and g as follows:

Thus, the standard error of Î1(backpackers), expressed as a percentage of

Î1(backpackers), amounts to

Similarly,

v a ˆr 2 s I ˆ 

1 ( backpackers) = 

+ 

n 1 M 2 

N 1 

ä 

ã å å 1 

m − 1 

M 

ë 

í ì ì 

n 

1 

3 

k = 1 

m 

3 

j = 1 

DI 2 

1 , k , j ( backpackers) 

n 1 á m − 1 é 

= 181. 5 + 20. 0 = 201. 5 .

á 201. 5 / 145. 0 é 100% = 9 . 8 % . 

v a ˆr 2 s I ˆ 

2 ( backpackers) = 

ä 

ã å å 1 

3 − 1 

54

ë 

í ì ì 3050

2 + á 3 é á 100 é 

54

ä 

ã å å 1 

2 − 1 

10

ë 

í ì ì 42. 5 

3 á 1 é

= 480. 1 + 31. 5 = 511. 6 .

ä 

ã å å 1 

n 1 

− 1 

N 1 

ë 

í ì ì 

n 1 

3 

k = 1 

D ˆ I 2 

1 , k ( backpackers) 

n 1 − 1 

ä 

ã å å 1 

4 − 1 

126

ë 

í ì ì 2250

3 + á 4 é á 100 é 

126

ä 

ã å å 1 

2 − 1 

10

ë 

í ì ì 63

4 á 1 é = 
23

The standard error of Î

2

(backpackers), expressed as a percentage of Î2(backpackers),

amounts to

Turning attention to the estimated variance of the estimates of recreational use by

bikers, we get

The standard error of Î1

(bikers), expressed as a percentage of Î

1

(bikers), amounts to

Similarly,

The standard error of Î2

(bikers), expressed as a percentage of Î

2

(bikers), amounts to

Lastly for backpackers,

Tˆ

1

(backpackers) = 126 × 145 = 18,270 per season during the week,

and

Tˆ

2

(backpackers) = 54 × 295 = 15,930 per season during the weekend,

á 511. 6 / 295. 0 é 100% = 7 . 7 % . 

v a ˆr 2 s I ˆ 

1 ( bikers) = 

ä 

ã å å 1 

n 1 

− 1 

N 1 

ë 

í ì ì 

n 1 

3 

k = 1 

D ˆ I 2 

1 , k ( bikers) 

n 1 − 1 

+ 

n 1 M 2 

N 1 

ä 

ã å å 1 

m − 1 

M 

ë 

í ì ì 

n 1 

3 

k = 1 

m 

3 

j = 1 

DI 2 

1 , k , j ( bikers) 

n 1 á m − 1 é 

= 

ä 

ã å å 1 

4 − 1 

126

ë 

í ì ì 618. 75

3 + á 4 é á 100 é 126

ä 

ã å å 1 

2 − 1 

10

ë 

í ì ì 11. 5 

4 á 1 é

= 49. 9 + 3 . 7 = 53. 6 .

á 53. 6 / 86. 25 é 100% = 8 . 5 % . 

v a ˆr 2 s I ˆ 

2 ( bikers) = 

ä 

ã å å 1 

3 − 1 

54

ë 

í ì ì 2450

2 + á 3 é á 100 é 

54

ä 

ã å å 1 

2 − 1 

10

ë 

í ì ì 15. 0 

2 á 1 é

= 385. 6 + 11. 1 = 396. 7 .

á 396. 7 / 150. 0 é 100% = 13. 3 % . 
24

and

T(backpackers) = 18,270 + 15,930 = 34,200 per season, ˆ

with a daily average of

Î(backpackers) = 34,200/180 = 190 per day during the season.

For bikers, the corresponding estimates are as follows:

Tˆ

1

(bikers) = 126 × 86.25 = 10,868 per season during the week,

and

Tˆ

2

(bikers) = 54 × 150 = 8,100 per season during the weekend,

and

T(bikers) = 10,860 + 8,100 = 18,968 per season, ˆ

with a daily average of

Î(bikers) = 18,968/180 = 105 per day during the season.

For camping at sites where access is controlled, it is not likely that any form of sampling will be required. We have in mind, for example, a campground with a staffed

access point or gate where registration occurs and fees are paid upon entry. For such

a site, recreational use statistics can be compiled from the registration information.

For overnight areas where access is neither controlled nor monitored, we propose that

campsites within the campground of interest be sampled. It is not sufficient to observe

entry through the access points to the overnight recreation area because entry on a

particular day may represent multiple days of recreation. For our objective of estimating daily recreational use, it is necessary to sample and observe that use among the

campsites on the selected days of sampling.

Sampling should be conducted at a time of day when campers are most likely to be at

the campsite and awake. We suggest early morning or just before sunset as the best

times. Once the time of day for campsite sampling is determined, it should be used

consistently throughout the sampling effort.

Sampling Strategy

for Overnight

Recreational Use

Areas
25

It is unlikely that the number of individuals in each group of overnight recreationists can

be observed without risking substantial measurement error in some cases. We therefore propose switching the focus of estimation from the number of recreating individuals

to the number of groups of overnight recreationists.

If the number of campsites is small enough to permit all to be observed on a selected

day of sampling, then the estimation formulas provided for transient-recreational-use

sites with single access points are applicable. To be specific, we tailor and present

those formulas here.

As established earlier, N, N1

, and N2, represent, respectively, the number of days in the

season of interest, in the subset of the season encompassing weekdays, and in the

subset encompassing weekend days. Similarly, n, n1, and n2 represent the number of

sample days.

To distinguish among the number of groups tallied on the different sample days in

stratum 1, we introduce this notation:

G1,1 = observed number of groups on sample day 1 in stratum 1

G1,k = observed number of groups on sample day k in stratum 1

G1,n1= observed number of groups on sample day n1 in stratum 1

Similarly for the tallies of groups in stratum 2, we have the following:

G2,1 = observed number of groups on sample day 1 in stratum 2

G2,k = observed number of groups on sample day k in stratum 2

G2,n2= observed number of groups on sample day n2 in stratum 2

Estimating the average number of groups per day by stratum—An estimate of the

average number of groups per day in stratum 1 is given by,

so that an estimate of the total number of groups using this site throughout the season

in stratum 1 is this average multiplied by the number of days in the stratum,

Small Campgrounds

G ˆ 

1 = 1 

n 1 

 

n 

1 

3 

k = 1 

G 1 , k = 

G 1 , l + ˇ + G 1 , k + ˇ + G 1 , n 1 

n 1 

 , 

T ˆ 

1 = N1 H G ˆ 1 .
26

An estimate of the average number of groups per day in stratum 2 is given by,

and the estimated total number of groups in stratum 2 throughout the season is,

Estimating the average number of groups per day for both strata—Because the

total number of groups in both strata is the sum,

the average number of groups in both strata is estimated as,

Variance estimation and confidence intervals—An estimate of the variance of Gˆ 1

depends on the extent to which the individual daily tallies, G1,1, G1,2,...,G1,n1vary around

Gˆ 1. As a computational aid, we suggest that these deviations be computed separately:

DG1,1 = G1,1 – Gˆ 1

DG1,k = G1,k – Gˆ 1

DG1,n = G1,n1 – Gˆ 1

The estimated variance of Gˆ 1 is designated as vâr( Gˆ

1

) and can be computed by the

following formula:

G ˆ 

2 = 1 

n 2 

 

n 

2 

3 

k = 1 

G 2 , k = 

G 2 , l + ˇ + G 2 , k + ˇ + G 2 , n 2 

n 2 

 , 

T ˆ 

2 = N2 H G ˆ 2 .

T ˆ = T ˆ 

1 + T ˆ 2 , 

G ˆ = T ˆ / N .

v a ˆr á G ˆ 

1 é = 

ä 

ã å å 1 

n 1 

 − 1 

N 1 

ë 

í ì ì 

n 1 

3 

k = 1 

DG2 

1 , k 

n 1 − 1 . 
27

For the weekend stratum,

where DG2,k = G2,k – Gˆ 2

, and so on.

For the estimate, Gˆ , pertaining to both strata combined,

Confidence intervals for average number of groups of recreationists visiting a

site—An approximate 90-percent confidence interval for G1

 is constructed as,

where t

df

0.90 is the 90th percentile point from the Student's t distribution with df = n1

 - 1

degrees of freedom, and where,

For the corresponding estimate of the seasonal total number of groups of recreationists

in stratum 1,

An approximate 90-percent confidence interval for G2 is constructed as,

v a ˆr á G ˆ 

2 é = 

ä 

ã å å 1 

n 2 

 − 1 

N 2 

ë 

í ì ì 

n 

2 

3 

k = 1 

DG2 

2 , k 

n 2 − 1 , 

v a ˆr á G ˆ é = 1 

N 2 N 2 

1 H v a ˆr á G ˆ 

1 é + N 2 

2 H v a ˆr á G ˆ 

2 é 

= 1 

N 2 v a ˆr á T ˆ 

1 é + v a ˆr á T ˆ 2 é . 

G ˆ 

1 " t 0 . 9 0

df v a ˆr á G ˆ 

1 é or G ˆ 1 " E% , 

E % = á t 0 . 9 0

df v a ˆr á G ˆ 

1 é / G ˆ 1 é 100% .

G ˆ 

2 " t 0 . 9 0

df v a ˆr á G ˆ 

2 é or G ˆ 2 " E% 

T ˆ 

1 " N1 H t 0 . 9 0

df v a ˆr á G ˆ 

1 é or T ˆ 1 " E% .
28

where t

df

0.90 is the 90th percentile point from the Student's t distribution with df = n2

 - 1

degrees of freedom, and where,

For the corresponding estimate of the seasonal total number of groups of recreationists

in stratum 2,

An approximate 90-percent confidence interval for G is constructed as,

where t

df

0.90 is the 90th percentile point from the Student's t distribution with df = n1

 + n2 - 2

degrees of freedom, and where,

For the corresponding estimate of the seasonal total number of groups of recreationists

in stratum 2,

Example 5—In this example we suppose that the small Pine River campground has

been sampled on n1 = 6 weekdays and n2 = 4 weekend days. The observed number of

groups of campers for each of the days is shown in table 7. For this example, as in the

previous ones, we suppose that the season of interest comprises N =180 days, of which

N1 = 126 are weekdays and the remainder, N2 = 54, are weekend days.

Table 7—Number of groups of campers

observed on each sampling day at the

Pine River campground

Stratum 1: weekdays Stratum 2: weekends

18, 23, 17, 19, 25, 26 31, 29, 26, 34

E % = á t 0 . 9 0

df v a ˆr á G ˆ 

2 é / G ˆ 2 é 100% .

T ˆ 

2 " N2 H t 0 . 9 0

df v a ˆr á G ˆ 

2 é or T ˆ 2 " E% . 

G ˆ " t 0 . 9 0

df v a ˆr á G ˆ é or G ˆ " E% , 

E % = á t 0 . 9 0

df v a ˆr á G ˆ é / G ˆ é 100% .

T ˆ " N H t 0 . 9 0

df v a ˆr á G ˆ é or T ˆ " E% . 
29

Based on these observations,

and

The variance of Gˆ 1 is estimated by,

and that of Gˆ 2

 by,

Using a value of t

5

0.90 = 2.015 a 90-percent confidence interval for G1

 is constructed as,

and with t

2

0.90 = 2.353, a 90-percent confidence interval for G2

 is constructed as,

Seasonal totals are estimated by,

G ˆ 

1 = 

1 

6 ( 18 + 23 + 17 + 19 + 25 + 26) = 21. 3  ̄ 

G ˆ 

2 = 

1 

4 ( 31 + 29 + 26 + 34) = 30. 0 .

v a ˆr á G ˆ 

1 é = 

ä 

ã å å 1 

n 1 

 − 1 

N 1 

ë 

í ì ì 

n 

1 

3 

k = 1 

DG2 

1 , k 

n 1 − 1 = 

ä 

ã å å 1 

6 − 1 

126

ë 

í ì ì 73. 3  ̄ 

5 = 2 . 33 , 

v a ˆr á G ˆ 

2 é = 

ä 

ã å å 1 

n 2 

 − 1 

N 2 

ë 

í ì ì 

n 2 

3 

k = 1 

DG2 

2 , k 

n 2 − 1 = 

ä 

ã å å 1 

4 − 1 

54

ë 

í ì ì 34. 0 

3 = 2 . 62 . 

21. 3  ̄ " 2 . 015 2 . 33 = 21. 3  ̄ " 3 . 08 = 21. 3  ̄ " 14. 4 % , 

30. 0 " 2 . 353 2 . 62 = 30. 0 " 3 . 81 = 30. 0 " 12. 7 % . 

T ˆ 

1 = N1 H G ˆ 1 = 126 H 21. 3  ̄ = 2688
30

and

A 90-percent confidence interval for T1

 is constructed as,

and a 90-percent confidence interval for T2 is constructed as,

Large campgrounds are those that have many campsites or the campsites are too

dispersed to permit observation of all of them on a selected day of sampling. Consequently, a subsample of campsites is selected and observed.

Under this two-stage sampling setup, suppose that,

S = the number of campsites in the site ,

s = the number of campsites sampled ,

and that on each day of sampling the fraction ,

h = s/S ,

of campsites are observed.

The estimation formulas mirror those given for two-stage sampling of access points for

transient-recreational-use sites. We have tailored them to the current problem as

presented below.

Let

G1,1,1 = observed number of groups on sample day 1 in stratum 1 at the first of the

subsampled campsites

G1,1,j = observed number of groups on sample day 1 in stratum 1 at the jth of the

subsampled campsites

G1,1,s = observed number of groups on sample day 1 in stratum 1 at the sth of the

subsampled campsites

T ˆ 2 = N2 H G ˆ 2 = 54 H 30. 0 = 1620 .

2688 " 2 . 015 36991 = 2688 " 387. 5 = 2688 " 14. 4 % , 

1620 " 2 . 353 7650 = 1620 " 205. 8 = 1620 " 12. 7 % . 

Large Campgrounds
31

Similarly for the tallies of groups in stratum 2, we have,

G2,1,1 = observed number of groups on sample day 1 in stratum 2 at the first of the

subsampled campsites

G2,1,j = observed number of groups on sample day 1 in stratum 2 at the jth of the

subsampled campsites

G2,1,s = observed number of groups on sample day 1 in stratum 2 at the sth of the

subsampled campsites

Let Gˆ

1,k represent the estimated average number of groups for the kth sampling day in

stratum 1, which is computed as,

Similarly for stratum 2,

The two-stage estimation formulas for Gˆ 1 and Gˆ 2 are

and

Formulas for Tˆ

1

, Tˆ

2

, G, ˆ and Tˆ remain unaltered from those given in the preceding

section.

The variance of Gˆ 1 and Gˆ 2 comes from two sources: there is the first-stage component

of variance that results from observing recreational use on only a fraction, n of N, of all

the days of the season; and there is the second-stage component of variance that

results from observing recreational use at only a fraction, h, of all the S campsites on

the site on any given sampling day.

G ˆ 

1 , k = 

1 

h 

s 

3 

j = 1 

G 1 , k , j .

G ˆ 

2 , k = 

1 

h 

s 

3 

j = 1 

G 2 , k , j .

G ˆ 

1 = 1 

n 1 

n 1 

3 

k = 1 

G ˆ 

1 , k 

G ˆ 

2 = 1 

n 2 

n 

2 

3 

k = 1 

G ˆ 

2 , k .
32

It is convenient to compute deviations from the average. In particular, suppose the

deviation of G1,k,j from the estimated average number of groups using the site on the kth

day of sampling is expressed as,

Because there are s campsites in the subsample and n1

 sample days in the first stage

of sampling, there are s × n1 deviations of the form DG1,k, j.

Additionally, suppose the deviation of Gˆ 1,k from the estimated seasonal average, Gˆ 1, is

expressed as,

Because there are n1 sample days, there are n1 deviations of the form DGˆ 1,k.

Similarly, the deviations in the weekend stratum, stratum 2, are designated as,

and

With these, the variance of Gˆ

1

 is estimated as,

Likewise,

D G ˆ 

1 , k = G1 , k − G ˆ 1 .

DG2 , k , j = G2 , k , j − l 

s 

s 

3 

i = 1 

G 2 , k , j , 

D G ˆ 

2 , k = G2 , k − G ˆ 2 .

v a ˆr 2 s á G ˆ 

1 é = 

ä 

ã å å 1 

n 1 

− 1 

N 1 

ë 

í ì ì 

n 1 

3 

k = 1 

D G ˆ 2 

1 , k 

n 1 − 1 + 

n 1 S 2 

N 1 

ä 

ã å å 1 

s − 1 

S 

ë 

í ì ì 

n 1 

3 

k = 1 

s 

3 

j = 1 

DG2 

1 , k , j 

n 1 á s − 1 é .

v a ˆr 2 s á G ˆ 

2 é = 

ä 

ã å å 1 

n 2 

− 1 

N 2 

ë 

í ì ì 

n 2 

3 

k = 1 

D G ˆ 2 

2 , k 

n 2 − 1 + 

n 2 S 2 

N 2 

ä 

ã å å 1 

s − 1 

S 

ë 

í ì ì 

n 

2 

3 

k = 1 

s 

3 

j = 1 

DG2 

2 , k , j 

n 2 á s − 1 é .

DG1 , k , j = G1 , k , j − l 

s 

s 

3 

i = 1 

G 1 , k , i .
33

As before,

As mentioned earlier, construction of confidence intervals for population parameters

based on estimates from two-stage and more complex designs is problematic. Any

useful treatment of this topic is outside the scope of this report.

We have repeatedly expressed confidence intervals in the alternate form,

estimated quantity + percentage,

where percentage is the length of the interval on either side of the estimated quantity

and expressed as a percentage of the estimated quantity. For example, from the

“Single Access Sites” subsection under “Sampling Strategy for Transient-RecreationalUse Areas,” section, we showed the 90-percent confidence interval for the average

number of individual backpackers on weekdays as,

where,

T ˆ 

1 = N1 H G ˆ 1 

v a ˆr áT ˆ 

1 é = N2 

1 H va ˆr áI ˆ 

1 é

T ˆ 

2 = N2 H v a ˆr áI ˆ 2 é

v a ˆr áT ˆ 

2 é = N2 

2 H va ˆr áI ˆ 

2 é

T ˆ = T ˆ 

1 + T ˆ 2 

v a ˆr áT ˆ é = v a ˆr áT ˆ 

1 é + v a ˆr áT ˆ 2 é

G ˆ = T ˆ / N 

v a ˆr áG ˆ é = v a ˆr áT ˆ é / N 2 

Planning a Sample of

Recreational Use

I ˆ 

1 ( backpackers) " E % , 

E % = t 0 . 9 0

df v a ˆr I ˆ 

1 ( backpackers) 

I ˆ 

1 ( backpackers) 

 100% .
34

One reason for preferring intervals expressed in this fashion is that many people can

better assimilate the information when it is in relative terms (as a percentage of the

estimated value) rather than absolute terms. Another reason is the natural link this

expression provides to the formula used to estimate the sample size needed in the

future.

If a past sample brought us to within ± 12 percent of the target, one might wish to take

a larger sample next time to achieve a “margin of error” of only 10 percent. Or the

decision could be made to conserve resources and select a smaller sample next time,

but one that will have a good chance of achieving a “margin of error” of at least 15

percent, say. These types of comparisons presume that the level of confidence, here

90 percent, remains constant.

Generally speaking, if E% continues to represent the “margin of error” achieved from a

sample, then E*% will denote the “margin of error” desired next time. Many sources

refer to E*% as an “allowable error,” where the implication is that a sample is desired

that is sufficiently large to allow the estimate to be within E*% of the targeted value at

least (1−α) 100 percent of the time, where (1−α) 100 percent is the desired confidence

level.

For sample-size planning, it stands to reason that the smaller value of E*% stipulated,

the larger the sample needed to achieve this level of precision.

The other factor (other than the desired confidence level and E*%) affecting the size of

the sample needed is the variation observed among the sampled values. More heterogeneous populations will exhibit greater variation among observations, and more

homogeneous populations will exhibit less variation.

The variance among population values is not at all influenced by sampling, as it is a

feature of the population itself. Conversely, the variance among population values

does affect the size of the sample needed to achieve a stipulated margin of error, E*%.

A well-known result, that many will find intuitively reasonable, is that the greater the

heterogeneity of the population, the larger the sample that will be needed to achieve

the desired margin of error.

Data from prior samples can be used to estimate this among-observations variation on

a relative or percentage basis with a statistic known as the coefficient of variation.

Again referring back to the estimation of Î1

(backpackers) from the “Single Access Sites”

section, the “coefficient of variation of I1

(backpackers)” can be estimated by,

CV1 % = 

1 

n 1 − 1 

n 1 

3 

k = 1 

DI 2 

1 , k ( backpackers) 

I ˆ 

1 ( backpackers) 

 100% .
35

Similarly, the coefficient of variation of the average daily number of individual backpackers on weekends can be estimated by,

The stipulated margin of error, E*%, and the estimated coefficient of variation can be

used jointly to estimate the size of the samples, say n1* and n2*, that will be needed to

achieve E*%. As an initial estimate of n1*, the following simplified formula is sufficient:

For this initial calculation, it is sufficient to choose a t-value of 1.645. Or if a 95-percent

confidence level is preferred, choose a t-value of 1.96.

On the basis of the initial estimate, reestimate n1* by using the following formula with a

value for t

df

0.90 corresponding roughly to n1

* − 1 degrees of freedom:

An estimate of n2

* can be computed in the same way.

These estimates of n1* and n2* need not be fine tuned any further. The reason for this

cautionary remark is that even with the best of planning and execution of the sample,

there is no guarantee that a sample of size n1* (or n2*) will necessarily achieve the

desired E*% margin of error, except in a probabilistic sense. There is little sense in

fretting over alternative values of n1* or n2* that differ by one, two, or similar inconsequential values.

In situations where no prior survey information exists, a small pilot survey can be

implemented to get a rough idea of the coefficient of variation values. Barring this, an

educated guess can be substituted.

The preceding development shows the same stipulated margin error for both the

weekday and weekend strata. Depending on the relative importance of reliable estimates from the two strata, the RM may prefer to stipulate a different E*% value for

each stratum.

CV2 % = 

1 

n 2 − 1 

n 

2 

3 

k = 1 

DI 2 

2 , k ( backpackers) 

I ˆ 

2 ( backpackers) 

 100% .

n ∗ 

1 = ä 

ã å å å å E ∗ % 

t 0 . 9 0

df H CV1 % 

ë 

í ì ì ì ì 

2 

 + 1 

N 1 

− 1 

 .

n ∗ 

1 = 

ä 

ã å å å å t 0 . 9 0

df H CV1 % 

E ∗ % 

ë 

í ì ì ì ì 

2 

 .
36

Example 6—Needed samples sizes for backpackers to achieve a E*% = 10 percent

margin of error at the same 90-percent confidence level:

Using the information from examples 1 and 2 as a source of prior information about the

natural variation of backpacking recreation on the Spruce Run Trail, one estimates

CV1

% = 16.6% and CV2% = 11.8%.

Thus as an initial estimate of the sample size in stratum 1 needed to achieve a E*% =

10 percent margin of error is computed as,

Adjusting the t-value so that it is appropriate for roughly 6 degrees of freedom (t

6

0.90 =

1.943) one computes,

An initial estimate of the sample size in stratum 2 needed to achieve a E*% = 10

percent margin of error is computed as,

Adjusting the t-value so that it is appropriate for roughly 3 degrees of freedom (t

3

0.90 =

2.353) one computes,

Shown below is a parallel set of computations for bikers, designed to estimate the

needed sample size to achieve a E*% = 10 percent margin of error at the 90-percent

confidence level.

Using the information from examples 1 and 2 as a source of prior information about the

natural variation of biking recreation on the Spruce Run Trail, one estimates CV1

% =

20.4% percent and CV2% = 28.6% percent.

n ∗ 

1 ( backpackers) = 

ä 

ã å å 

1 . 645 H 16. 6 % 

10. 0 ∗ % 

ë 

í ì ì 

2 

 = 7 . 5 days .

n ∗ 

2 ( backpackers) = 

ä 

ã å å 1 . 645 H 11. 8 % 

10. 0 ∗ % 

ë 

í ì ì 

2 

 = 3 . 8 days .

n ∗ 

2 ( backpackers) = ä 

ã å å 10. 0 % 

2 . 353 H 11. 8 % 

ë 

í ì ì 

2 

 + 1 

54

− 1 

 = 6 . 7 ï 7 days .

n ∗ 

1 ( backpackers) = ä 

ã å å 10. 0 % 

1 . 943 H 16. 6 % 

ë 

í ì ì 

2 

 + 1 

126

− 1 

 = 9 . 6 ï 10 days .
37

Thus as an initial estimate of the sample size in stratum 1 needed to achieve a E*% =

10% margin of error is computed as,

Adjusting the t-value so that it is appropriate for roughly 11 degrees of freedom (t

11

0.90 =

1.818), one computes,

An initial estimate of the sample size in stratum 2 needed to achieve a E*% = 10

percent margin of error is computed as,

Adjusting the t-value so that it is appropriate for roughly 21 degrees of freedom (t

21

0.90 =

1.721) one computes,

Although the above approach to planning the needed sample size focuses on controlling the margin of error within each stratum, it does not control the overall margin of

error of the combined estimate: For example,

An alternate approach to sample size planning in a stratified random setup is to estimate the combined size of the stratified sample, say n*, and then to allocate n* to n1

*

and n2*, according to some plan. Cochran (1977: sec. 5.9) provides the details. The

drawback of this approach is that control is conceded over the margin of error one

wishes to achieve in the individual strata.

For stratified samples aimed at providing estimates of multiple characteristics, proportional allocation of the overall size of sample usually has much to recommend it. With

Complicating Factors

n ∗ 

1 ( bikers) = 

ä 

ã å å 1 . 645 H 20. 4 % 

10. 0 ∗ % 

ë 

í ì ì 

2 

 = 11. 3 days .

n ∗ 

1 ( bikers) = ä 

ã å å 10. 0 % 

1 . 818 H 20. 4 % 

ë 

í ì ì 

2 

 + 1 

126

− 1 

 = 12. 3 ï 12 days .

n ∗ 

2 ( bikers) = 

ä 

ã å å 

1 . 645 H 28. 6 % 

10. 0 ∗ % 

ë 

í ì ì 

2 

 = 22. 1 days .

n ∗ 

2 ( bikers) = ä 

ã å å 10. 0 % 

1 . 721 H 28. 6 % 

ë 

í ì ì 

2 

 + 1 

54

− 1 

 = 16. 7 ï 17 days .

I ˆ ( backpackers) = 

N 1 H I ˆ 

1 ( backpackers) + N 2 H I ˆ 2 ( backpackers) 

N .
38

proportional allocation, the size of the sample in each stratum is proportional to the size

of the stratum, for example, n1* ∝ N1 and so on. The conventional wisdom is that with

proportional allocation, none of the various characteristics will be estimated optimally,

but on the other hand, the characteristics will be sampled with a frequency proportional

to their occurrence in the population.

When sampling recreational use, it might be better to make the allocation proportional

to something other than the size of the stratum. Our rationale here is that the use of N1

and N2 to determine the allocation of the sample makes sense if the proportions N1/N

and N2/N are good indicators of the relative recreational use among strata. However, if

most of the recreational use occurs on weekends, a better planning tactic may be to set

n2*/n equal to an educated guess of the proportional recreational use on weekends,

instead of gearing it to the size of the weekend stratum as measured by the number of

days within it. The following example illustrates this tactic.

Example 7—Past experience suggests that 60 percent of the recreational use of a trail

by mountain bikers occurs on the weekend. In this case, a future survey of overall size

n* = 40 days may be allocated such that there are n2 = 24 days (0.6 × 40) of sampling on

weekends, and n1 = 16 days during the week, a tactic that shifts the sampling effort to

include more of the heavy use days than would be included under a conventional

proportional allocation.

For surveys in which multiple characteristics (for example, trail use estimates by

mountain bikers, hikers, overnight campers, by weekday and by weekend) are being

estimated, the planning of future sample sizes to achieve desired levels of precision is

problematic because the optimal sizes of samples within strata for one characteristic

will rarely match those of another characteristic, and indeed may be quite opposite.

Unless the RM mounts separate samples for each recreational use of interest, this

problem is insurmountable.

It is evident that for any recreation survey encompassing not only multiple strata but

also multiple recreational uses, planning of the future sample is far from straightforward. The essential problem is that it may not be possible to stipulate an overall

sample size adequate to provide sufficiently precise estimates of multiple characteristics. Compromises will have to be made to avoid being swamped in a maze of costly,

unipurpose surveys. The terms of this compromise cannot be determined statistically.

An oft-cited solution is to base the sample size on the requirements for precise estimation of the most important characteristic, and to let that implicitly determine the precision with which the remaining characteristics can be estimated.

The sampling strategies presented in this report surely will get adapted and refined as

experience with them is gained. All the sampling designs described in this report are

either one- or two-stage stratified random designs, which may strike some as quite

complicated already! Especially if the results of recreational-use surveys will be

subjected to public and official scrutiny, we urge that the sampling design stay comparatively simple, as these designs are. All these methods provide unbiased estimates

of recreational use, at least in principle. Measurement error is a biasing agent: hence,

our introductory caution about making extra efforts to minimize this source of inaccuracy. Measurement error notwithstanding, all the methods presented in this report are

free of the bias that is insinuated as a result of self-selection into the sample.

Epilogue
39

Cochran, W.G. 1977. Sampling techniques. New York: Wiley. 428 p.

Sample sizes, n*, needed to achieve E*% margin of error at various levels of CV%:

CV%

E*% 10 25 50 100

5 13 70 271 1082

10 9 19 70 271

15 18 10 32 122

20 10 9 19 70

Literature Cited

Appendix

Having championed simplicity, we should admit that it is easy to envision specialpurpose surveys of a particular recreational use that could be made much more

efficient by a more complex design or estimator. For example, past information or

auxiliary information on file might be used to select the sample days differentially

depending on anticipated use, or the information might be incorporated into a ratio or

regression estimator. For such special-purpose surveys, something other than the

simple strategies proposed here may be justifiable. Still, we reemphasize our introductory remark that for multiuse surveys, simplicity of design is a virtue worth striving for.

We have attempted to provide a set of basic tools applicable to a broad range of

recreational sites to estimate numerous recreational uses. Certainly there will be

combinations of sites and uses for which these methods are insufficient. As experience

with these mavericks is accumulated, so too will our knowledge of how to design

strategies to meet the two-fold objective of simplicity and efficiency.
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Gregoire, Timothy G.; Buhyoff, Gregory J. 1999. Sampling and estimating

recreational use. PNW-GTR-456. Portland, OR: U.S. Department of

Agriculture, Forest Service, Pacific Northwest Research Station. 39 p.

Probability sampling methods applicable to estimate recreational use are

presented. Both single- and multiple-access recreation sites are considered.

One- and two-stage sampling methods are presented. Estimation of recreational use is presented in a series of examples.

Keywords: Sampling strategies, confidence intervals, stratified sampling.

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the States and private forest owners, and management

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it strives—as directed by Congress—to provide

increasingly greater service to a growing Nation.

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