Sampling the Evidence of Extension Program Impact^{1}
Evaluating cooperative Extension programs is a process which includes gathering evidence about program outcomes and impacts. One part of this process is the determination of how much data is necessary to show whether or not a program had the intended outcome. For example, if a program on the adoption of a new technology by farmers is being evaluated, should each and every farmer be asked if he or she adopted the technology or should a sample of farmers be asked the question?
A sample can provide an appropriate amount of evidence for an evaluation. A sample can also save the valuable time, money, and labor of Extension professionals. Time is saved because fewer people, farmers, 4Hers, etc., must be interviewed or surveyed; thus the complete set of data can be collected quickly. Money and labor are saved because less data must be collected. In addition, errors from handling the data (e.g. entering data into a computer file) are likely to be reduced because there are fewer opportunities to make mistakes.
The purpose of this publication is to provide an overview of sampling procedures for obtaining data to evaluate Extension programs. Strategies for selecting a sample will be reviewed. A second publication, Determining Sample Size, PEOD6, should also be consulted (https://edis.ifas.ufl.edu/pd006).
The Evaluation Purpose
The first step in determining the sampling procedures to be used in an evaluation is a clear statement of the research or evaluation question. Ask yourself, What do I want to know?
Have the felt needs of residents who live in Manatee County been reduced?
What practices did farmers in Columbia and Suwannee Counties adopt as a result of the Farming Systems Research and Extension program?
Has income among households with a new homebased business increased more than those without one?
The above questions suggest that the purpose of the evaluation can vary. The purpose may be as simple as documenting the change of indicator variables (that program activities are assumed to affect), or the purpose may include a more rigorous analysis that compares changes by program participants with changes by nonparticipants in order to estimate the impact that can be attributed to the program. This type of question has important implications for the sample selection process.
Defining the Population
A good problem statement is necessary to identify the population relevant to evaluating program impacts. The population is composed of the individuals or groups that are affected by the Extension program and thus are the focus of the evaluation. The residents of Manatee County or small farmers in Suwannee and Columbia counties are examples of populations for the evaluation questions stated above. The individual residents or farmers in these examples are called sampling units or elements.
The population can be defined by geographical, demographic, economic, and social characteristics, as well as by the content of the survey (Ilvento et al., 1986). These characteristics include county of residence, age, sex, race, marital status, income, household size, farm size, and so on. A time frame can also be used to specify the population. For example, a population may include only people who have participated in a program during the last six months.
Defining a population too narrowly can make it difficult, if not impossible, to obtain a list of the individual elements (Sudman, 1976). For example, a list of peanut farmers who are 18 to 45 years old and work offfarm jobs is unlikely to exist.
Sometimes the source of the data is not the same as the sampling unit or element (Sudman, 1976). In the third example of an evaluation question shown earlier, the sampling element is the household but the data is obtained from individuals, e.g., the head of household. Similarly, evaluations of programs involving youth often sample adults, such as parents or teachers, to report their observations about what youth learn or do.
To Sample or Not to Sample
With the purpose of the evaluation stated and the population defined, the decision of whether to use a sample or a complete census (in which everyone in the population is included) can be made. There are several considerations to take into account. First, is collecting data on all the elements in the population feasible? If the cost and time requirements are prohibitive, a sample may be the only alternative. This is likely the case for a mass media evaluation survey in counties with large populations, e.g., MiamiDade County, Florida, which has over 2 million residents. Collecting data on a large number of individuals can also increase errors from data handling because the large volume creates more opportunities for error. On the other hand, if an evaluator wants to survey the 150 farmers in a recordskeeping program, the advantages of sampling are less clear. A complete census of the 150 farmers may be the better alternative because error due to sampling is eliminated. A census also has the advantage of providing information on each and every individual in the population of the program.
The choice between a census and a sample also depends on the scope of the evaluation. A census can be a quick and efficient method if an agent or specialist wants to determine the extent of learning or practice change among the 150 farmers in the recordskeeping program. For a more rigorous impact study, a sample of all the farmers in the county or area and not just those in the program is more appropriate. A sample of the wider population allows the comparison of adoption rates between farmers who are involved in Extension programs and those who are not. This idea applies to Extension programs in other areas as well.
Nonprobability and Probability Samples
Suppose you have decided to use a sample rather than a census. Should you use a nonprobability or a probability sample? Nonprobability samples use procedures for selection that are not based on chance. With this type of sample, there is no way to accurately estimate the chance of any element being selected. The quality of a nonprobability sample depends on the knowledge, judgment, and expertise of the researcher. At the same time, nonprobability samples can be quite convenient and economical.
Nonprobability samples include haphazard, convenience, quota, and purposive samples. Haphazard samples are those in which no conscious planning or consistent procedures are employed to select sample units (Cochran, 1963).
Convenience samples are those in which a unit is selfselected (e.g., volunteers) or easily accessible. Reaction surveys at the end of an Extension program, in which the respondents selfselect to participate, are an example of a convenience sample. Although this type of sample can yield useful information, these samples must be used with caution in inferring impacts of a program.
Quota samples are those in which a predetermined number of units with certain characteristics are selected. A sample of 50 men and 50 women to be interviewed on a busy street is an example of this type. The quality of the sample depends on the evaluator's ability to determine the relevant characteristics and size of the quota.
Researchers select units (e.g., individuals) for a purposive sample on the basis of characteristics or attributes that are important to the evaluation (Smith, 1983). The units used in a purposive sample are sometimes extreme or critical units. Suppose we are evaluating the adoption rates of a technology by farmers and we want to know if large farmers differ from small farmers. A sample of extreme units, e.g., farms of 1,000 or more acres and farms of 100 or less acres, would provide information to make this comparison. Similarly, if we want to evaluate why people adopt water conservation practices, households who have decreased their water consumption by 25% could be considered critical units for a sample. A small purposive sample can also be used to pretest the survey instrument of a larger sample (Sudman, 1976). Similarly, a pretest using a sample of critical units (e.g., experts or targeted clients) can identify problem questions, and these can be corrected before the larger survey is implemented.
A probability sample is one in which every element in the population has a known, nonzero probability of selection (Sudman, 1976, p. 49). Because the probability is known, the sample's statistics can be generalized to the population at large (at least within a given level of precision). These statistics include means, proportions, and regression parameters. There are several types of probability samples, e.g., simple random samples and stratified samples. The procedures to select the sample are described below. Probability samples generally are preferred over nonprobability samples because the risk of incorrectly generalizing to the population is known.
The Sampling Frame
The sampling frame is a list of units or elements from which the sample is selected. The ideal frame lists every element separately, once and only once, and nothing else appears on the list (Kish, 1965). In many cases, the list does not contain exactly the same elements as the population from which information is desired. In addition, older lists are likely to be less accurate than more recently compiled lists. The rectangle (areas A and B) in Figure 1, represents the population of interest (e.g., households, citrus growers, 4H groups, etc.). The areas A and C represent the sampling frame or list. As shown in the figure, some elements of the population are missing from the list (area B), while there are elements contained on the list which are not a part of the population (area C). The latter are "foreign" elements, such as livestock farmers listed along with citrus growers, or duplicate listings.
Each list that is used as the sampling frame should be screened for duplicates and, when possible, foreign elements^{1}. In addition, some estimate of the number of elements that are missing from the list should be made (this is called coverage error; see Dillman et al., 2014).
If too many elements are missing, the sample will not be representative of the population in which we are interested. One alternative is to look for another list to use as the sampling frame.
Leslie Kish (1965) identified four common problems of sampling frames or lists:
missing elements, noncoverage, or an incomplete frame
blanks or foreign elements
duplicate listings
clusters of elements combined into one listing
The first three were discussed above. The fourth, clusters of elements, refers to situations where individuals are not listed separately, e.g., members of a household. This is only a problem if we are interested in the responses of each of the individuals rather than the household as a whole.
According to Kish (1965), there are three responses to these problems. First, the problem can be ignored or disregarded. This response may be appropriate if the problem is relatively minor in comparison to other sources of error (such as inaccurate data from poorly worded questions) and correcting the list is costly and timeconsuming. Second, the population can be redefined to fit the sampling frame. Let's assume that we are studying citrus growers in Lake County and the list of growers from the county office is incomplete. In this case, the study population would be redefined as citrus growers known to Extension in Lake County. We can use that list if the research is not seriously deflected from its purpose. Third, we can spend the time and effort to correct the list.
If one of the three responses are not feasible, one of the following remedies for the four types of frame problems can be applied (see Kish, 1965):
Missing elements. To identify or survey elements missing from the list, a supplement in a separate stratum (sample grouping) can be employed. The Bureau of the Census uses fieldworkers to count the homeless in addition to sending surveys to every household in the country. Similarly, a survey of citrus growers from the Extension list might be supplemented by fieldwork. This would include driving across the county and stopping at farms not on the citrus list.
Foreign elements. Omit foreign elements from the sample if they can be identified. If a probability sample is to be used, do not replace the element with the next one on the list because this changes the probability of selecting each individual element.
Duplicate listings. This problem can be addressed by selecting only the first, last, oldest, or newest listing. Any unique feature can be used to select one of the listings. If two or more lists are used, remove all the names from the second list that appear on the first. Whatever criterion is selected, it should be applied in a consistent manner for all duplicates.
Clusters of elements. One way to address the problem of clustering is to include all the elements within each selected listing, e.g., all the people living at the same address. A second method is to select one element at random from those in the selected listing and weight it by the number of elements in the listing.
These remedies are basic common sense techniques. The key idea here is to apply a consistent, explicit rationale for including and excluding elements on the list from which the sample is drawn.
There are a number of lists that can be used to draw samples. The usefulness of these varies with the purpose of the study and the type of sample. Some types of lists include:
lists of driver's licenses
lists of utility company users (telephone, electric, water, and sewage)
organizational directories or membership lists
lists from the tax collector or assessor (property owners)
lists of Extension clients/program attendees, community or organizational directories
addressbased samples using the US Postal Service's Delivery Sequence File
In recent years, addressbased samples have become popular for general population surveys at the state and national levels (Dillman et al., 2014). These lists are useful for needs assessment and surveys that assess exposure to mass media Extension programs. Lists of Extension clients or organizational directories can be used to assess program impact for specific groups, e.g., citrus growers, Master Gardeners, or 4H leaders.
There are occasions when a list is unavailable or insufficient for the study's purpose. One method to overcome this deficiency is to specify a procedure based on location or some other known characteristic. If a probability sample is desired, then the procedures must allow the elements to have an equal chance of selection. Cluster or area sampling is one procedure that does this. For example, if a sample of children ages 8 to 18 is desired, but no list is available, schools with children of those ages can be identified and randomly selected. Within each school that is selected, all the children 8 to 18 can then be surveyed (or a list of the children can be sampled).
Selecting a Probability Sample
After the sampling frame or list has been obtained and any corrections made, the procedures for selecting the sample from the population must be determined. If a probability sample is planned, there are several methods for selecting a sample.
Simple Random Sample
A simple random sample is one of the easiest and least complex samples to select. With this method, each element on the list has an equal probability of selection. Typically, each element on the list, e.g. the name of a farmer, is assigned a number. Then, those numbers selected from a table of random numbers or randomly generated by a computer program are included in the sample. A table of random numbers is easy to use for small samples but becomes cumbersome for large samples.
To use a table of random numbers, use the following procedures (cf. Sudman, 1976):
Assign a number to each name on the list. Each sampling element (person, household, farm, etc.) must be uniquely identified.
Select a starting point. You can begin anywhere in the table and move in any direction (see Table 1).
Determine the number of columns to read. If there are 10,000 elements in the population, you must use five columns of digits; if there are 300 elements in the population, then only three columns are needed.
Select numbers from the table. Suppose you are studying a population of 196 ping pong balls. You would then select any three digit number from 001 to 196. Any number over 196 is discarded because these numbers do not correspond to any element in the population. Now suppose you select 149. The ping pong ball which is numbered 149 on the list is selected for the sample.
Discard any duplicate numbers that you select. This means that you are sampling with replacement.
Select numbers until you obtain the desired sample size. Suppose we want a sample of 20 ping pong balls from our population of 196. We would continue drawing nineteen more numbers (in addition to 149) between 1 and 196 from the table of random numbers. One such sample included the following elements: 50, 6, 149, 178, 176, 55, 41, 94, 87, 29, 162, 11, 43, 120, 156, 119, 17, 180, 134, 169. Figure 2 illustrates this simple random sample of 20 (Note: The ping pong balls are numbered from the upper left corner [1] to the bottom right corner [196]).
Spreadsheet programs are a quick and easy alternative for selecting a simple random sample. Using the ping pong ball example, the RANDBETWEEN formula can be used in Excel to generate a random number between 1 and 196 for each element in the population (e.g., ping pong ball). After each ping pong ball is assigned a random number, then the numbers are sorted and the 20 smallest (or largest) random numbers are selected for the sample.
Although simple random samples are easy to select, they have one undesirable quality. On rare occasions, you can select a sample that is far off from the true population mean (Slonim, 1957). To illustrate, suppose that we select a sample of 20 from a population of 196 ping pong balls. There are thousands^{2} of possible samples of 20 from this population. If the sample we select happens to have the 20 ping pong balls with the smallest or largest level of what we are measuring, then the mean of the sample is likely to be quite different from the population mean.
One way to avoid getting an "extreme" sample is to use additional information about the population to create a stratified sample. This method is explained next.
Stratified Random Sample
To improve estimates of means or proportions obtained from a simple random sample, the population can be arranged into strata or groups. Age, sex, and race are some demographic characteristics that are commonly employed to stratify samples. Stratified random samples require you to obtain information about the population prior to the sampling process. The sampling frame, which lists the population, as well as this auxiliary data, previous samples, and research papers, are some of the sources of information that can be used to stratify samples (Ilvento et al., 1986). Stratified samples are usually more accurate than random samples because each group or strata is wellrepresented in the sample. Within each stratum, a separate sample is randomly selected.
Three types of stratified samples are commonly employed in surveys: proportionate, disproportionate or optimal allocation, and equal size samples. In proportionate stratified samples, the sample size in each strata is made proportionate to the population size of the stratum (Kish, 1965). For example, if 16% of the population in a program is 65 years of age or older, then 16% of the sample should contain people in that age group.
Optimal allocation employs formulas to determine the sample size of each strata or group that will maximize the precision of the statistics for a particular total sample size (Slonim, 1957). The basic idea behind optimal allocation is that larger samples are required for strata with a high degree of variability than for those with less variability in order to yield the same level of precision on the variable of interest. Optimal allocation can also be used to minimize the cost of data collection when the cost varies from stratum to stratum. For a given budget and level of accuracy, the sample size of each stratum is determined on the basis of the cost of collecting the data for the desired level of precision.
Equal size samples are stratified samples in which the sample size of each strata or group is the same. For example, if we sample 300 men and women from a population, we would select 150 men and 150 women. This type of sample is preferred when two or more groups are to be compared in the evaluation.
The size of stratified samples is governed by one additional consideration. If subgroups or strata are designated as domains of study, as well as the total sample, the sample proportion or size may have to be adjusted to yield a desired level of accuracy (Kish, 1965). For example, suppose we have a sample of 300 farmers and only 6% (18) are 65 years of age or older. If we wish to make any conclusions about the subgroup of farmers who are 65 or older, we would need a larger number of older farmers (the determination of sample size is discussed in PEOD6, Determining Sample Size, https://edis.ifas.ufl.edu/pd006). One way to do this is to sample 300 farmers but with 100 of those being 65 years of age or older. Oversampling of older farmers allows conclusions to be made for both the total sample and the subgroups.
The use of stratified random samples requires additional procedures for calculating sample statistics such as means and variances. According to Kish (1965, p.75), a separate stratum mean (or other statistic) is calculated and these are weighted to form a combined estimate for the entire population. In the above example of farmers, the mean of the 100 farmers who are 65 or older is multiplied by .06 (the proportion of this group in the population) and the mean of the remaining 200 farmers in the sample is multiplied by .94. These are added to obtain the mean for farmers of all ages. Note that weighting the mean is not required for proportionate samples. The variances are also computed separately and then weighted in forming the combined estimate for the population (this also is necessary for proportionate samples).
To illustrate the process of selecting a stratified random sample, refer to Figure 3. Suppose we have 49 each of orange, blue, yellow, and green ping pong balls (a total of 196) as represented by the four sections. A proportionate or equal size stratified sample would produce statistics with the same precision for each group (assuming equal variances), and for the total population. Thus, 5 ping pong balls are randomly selected for each color, as shown in Figure 3. If we selected the simple random sample shown in Figure 2, we would have selected 5 orange balls, 5 green balls, 3 blue balls and 7 yellow ping pong balls. The estimate of size or weight of yellow ping pong balls would be more precise than that for blue balls in the simple random sample because of the "extra" data on yellow balls.
Systematic Sample
Systematic samples are widely used and easy to implement.
A systematic sample selects the first element randomly and then every i^{th} element on the list afterwards. Suppose a sample of 20 ping pong balls is selected from a population of 196. The interval between selected elements from the list would be 20/196 or 10 (always round down to the nearest whole number to ensure enough elements are selected). The starting point would be a number between 1 and 10 that is selected from a table of random numbers. If 2 were chosen, the sample would include the 2nd, 12th, 22nd,... through the 192nd ping pong balls on the sampling list, as shown in Figure 4.
Systematic samples, like simple random samples, give each element an equal (but not independent) chance of being selected. This procedure can also be used if you do not have a list when the elements are arranged in space (such as houses along a road). However, if the arrangement of the population on the list (or road) has some pattern or periodicity, then the sample may become biased. For example, if a directory of couples always listed the man first, an interval that caused an odd number to always be selected would include only men in the sample. Because of the danger of bias, random numbers selected from a table of random numbers or those generated by computer (for larger samples) are to be preferred when a list is available.
Cluster or Area Samples
When a list of the entire population is nonexistent, hard to obtain, or the cost of surveying dispersed individuals is prohibitive, cluster sampling can facilitate the data collection process. Cluster sampling is a method of selecting sampling units in which the unit contains a cluster of elements (Kish, 1965, p.148). Some types of clusters include employees of business firms, children in schools, dwellings in city blocks, and residents in counties or states. The last two are geographical clusters or areas.
To illustrate, suppose we wish to evaluate a statewide program in energy conservation with a facetoface survey of 1,000 households. Although a sample could be drawn from the addresses in the Delivery Sequence File, the cost of surveying individuals who are dispersed throughout the state would be high. A cluster sample can reduce the survey cost and capture respondents in groups that are likely to be underrepresented. A cluster sample for this case could begin by randomly selecting a sample of counties in the state, then randomly selecting county subdivisions and neighborhoods, and finally randomly selecting street segments. Each household on a selected street segment would be interviewed. Note that the cluster sample in this example is composed of several stages. In addition, the probability of selecting a particular household is the product of the probability of selecting its street segment, neighborhood, town, and county (Kish, 1965).
Using the ping pong ball example, suppose we find that ping pong balls are sold in packages containing 4 balls. Thus, the population of 196 balls is distributed among 49 packages. To obtain the desired sample size of 20 balls, we first calculate the proportion of the population that the sample comprises (20/196 is about 10 percent). Next we multiply the number of packages (clusters) of ping pong balls by that proportion to determine the number of packages to be selected. This is 49 x .1, or 5 packages (here we round up to get the desired sample size). Finally, the 2nd, 13th, 19th, 26th and 39th packages of ping pong balls were randomly selected (see Figure 5). The probability of selecting any single ball is the same as that for selecting its package, i.e., 1 in 10.
Ideally, individual clusters in cluster samples should be as heterogeneous as possible. For example, each package should contain an orange, blue, green, and yellow ping pong ball. This is the reverse from stratified samples, where each strata is homogeneous. Recall that the five ping pong balls in each group of the stratified group random sample were the same color. In practice, clusters are often somewhat homogeneous, such as households located on the same street. Consequently, the sample results tend to be less precise than other techniques for the same size sample but more cost efficient (Slonim, 1957).
Selecting a Sample Design
The choice of a sample design will be largely determined by the amount of information that is available for the population. If characteristics of the population are known, then a stratified sample can be used to obtain more precise data. If little is known about the population, then a less complex design, such as simple random or systematic samples, can be used. When a list is unavailable or incomplete, a cluster sample may be the best choice. For large national or statewide surveys, these methods can also be combined, such as a stratified multistage cluster sample, to provide useful and costeffective samples.
Concluding Comments
The sampling process is multifaceted. A welldesigned sample can provide representative data which is useful for evaluating Extension programs. Such a sample begins with a consideration of the purpose of the evaluation, the characteristics and size of the population, the availability of an accurate and uptodate sampling frame, and the procedures for selecting who will be in the sample. Addressing these issues, along with determining the size of the sample, will contribute to a credible and rigorous evaluation.
Endnotes
Another way to identify foreign elements is through the use of screening questions on the survey instrument. This is especially useful in telephone surveys, where the interviewer can abbreviate the interview if the respondent does not meet the selection criteria and save valuable time and money.
The actual number of possible samples is 196! / 20! 176!. What this equals is too large to compute on my smartphone.
References
Cochran, W. G. (1963). Sampling Techniques, 2nd Ed., New York: John Wiley and Sons, Inc.
Dillman, D. A., J. D. Smyth, & L. M. Christian. (2014). Internet, phone, mail, and mixedmode surveys: The tailored design method. (4th ed.). Hoboken, NJ: John Wiley and Sons.
Ilvento, Thomas W., Paul D. Warner & Richard C. Maurer. (1986). Sampling Issues for Evaluations in the Cooperative Extension Service. Kentucky Cooperative Extension Service. University of Kentucky.
Kish, Leslie. (1965). Survey Sampling. New York: John Wiley and Sons, Inc.
Slonim, M. J. (1957). "Sampling in a Nutshell," American Statistical Association Journal. June.
Smith, M. F. (1983). Sampling Considerations In Evaluating Cooperative Extension Programs. Florida Cooperative Extension Service Bulletin PE1. Institute of Food and Agricultural Sciences. University of Florida.
Sudman, Seymour. (1976). Applied Sampling. New York: Academic Press.
Tables
Table of Random Digits.
(01) 
(02) 
(03) 
(04) 
(05) 
(06) 
(07) 
(08) 
(09) 
(10) 

(0001) 
9492 
4562 
4180 
5525 
7255 
1297 
9296 
1283 
6011 
0350 
(0002) 
1557 
0392 
8989 
6898 
3824 
6013 
0020 
8582 
5059 
9324 
(0003) 
0714 
5947 
2420 
6210 
3824 
2743 
4217 
3707 
5894 
0040 
(0004) 
0558 
8266 
4990 
8954 
7455 
6309 
9543 
1148 
0835 
0808 
(0005) 
1458 
8725 
3750 
3138 
2499 
6017 
7744 
0485 
3010 
9606 
(0006) 
5169 
6981 
4319 
3369 
9424 
4117 
7632 
5457 
0608 
4741 
(0007) 
0328 
5213 
1017 
5248 
8622 
6454 
8120 
4585 
3295 
0840 
(0008) 
2462 
2055 
9782 
4213 
3452 
9940 
8859 
1000 
6260 
2851 
(0009) 
8408 
8697 
3982 
8228 
7668 
8139 
3736 
4889 
7283 
7706 
(0010) 
1818 
5041 
9706 
4646 
3992 
4110 
4091 
7619 
1053 
4020 
(0011) 
1771 
8614 
8593 
0930 
2095 
5005 
6387 
4002 
7498 
0066 
(0012) 
7050 
1437 
6847 
4679 
9059 
4139 
6602 
6817 
9972 
5360 
(0013) 
5875 
2094 
0495 
3213 
5694 
5513 
3547 
9035 
7588 
5994 
(0014) 
2473 
2087 
4618 
1507 
4471 
9542 
7565 
2371 
3981 
0812 
(0015) 
1976 
1639 
4956 
9011 
8221 
4840 
4513 
5263 
8837 
5868 
(0016) 
4006 
4029 
7270 
8027 
7476 
7691 
6362 
1251 
9277 
5833 
(0017) 
2149 
8162 
0667 
0825 
7353 
4645 
3273 
1181 
8526 
1176 
(0018) 
1669 
7011 
6548 
5851 
8278 
9006 
8176 
1268 
7113 
4548 
(0019) 
7436 
5041 
4087 
1647 
7205 
3977 
4257 
9008 
3067 
7206 
(0020) 
2178 
3632 
5745 
2228 
1780 
6043 
9296 
4469 
8108 
5005 
(0021) 
1964 
3043 
3134 
8923 
1019 
8560 
5871 
7971 
2233 
7960 
(0022) 
5859 
7120 
9682 
0173 
2413 
8490 
6162 
1220 
3710 
5270 
(0023) 
2352 
1929 
5985 
3303 
9590 
6974 
5811 
4264 
0248 
4295 
(0024) 
9267 
0156 
9112 
2783 
2026 
0493 
9544 
8065 
4916 
3835 
(0025) 
4787 
0119 
1261 
5197 
0156 
2385 
9957 
0990 
6681 
2323 
(0026) 
5550 
0699 
8080 
1152 
6002 
2532 
3075 
2777 
8671 
4068 
(0027) 
7281 
9442 
4941 
1041 
0569 
4354 
8000 
3158 
9142 
5498 
(0028) 
1322 
7212 
3286 
2886 
9739 
5012 
0360 
5800 
9745 
8640 
(0029) 
5176 
2259 
2774 
3641 
3553 
2475 
1974 
4578 
3388 
6656 
(0030) 
2292 
1664 
1237 
2518 
0081 
8788 
8170 
5519 
0467 
4646 
(0031) 
6935 
8265 
3393 
4268 
4429 
1443 
4670 
4177 
7872 
9298 
(0032) 
8538 
5393 
8093 
7835 
0484 
2550 
0827 
3112 
1065 
0246 
(0033) 
4351 
0691 
0592 
2256 
4881 
4776 
4992 
2919 
3046 
3246 
(0034) 
6337 
8219 
9134 
9611 
8961 
4277 
6288 
2818 
1603 
4084 
(0035) 
2257 
1980 
5269 
9615 
8628 
4715 
6366 
1542 
7267 
8917 
(0036) 
8319 
9526 
0819 
0238 
7504 
1499 
8507 
9767 
1345 
7509 
(0037) 
1717 
8853 
2651 
9327 
7244 
0428 
6583 
2862 
1452 
8061 
(0038) 
6519 
9348 
1026 
4190 
4210 
6231 
0732 
7000 
9553 
6125 
(0039) 
1728 
2608 
6422 
6711 
1348 
6163 
4289 
6621 
0736 
4771 
(0040) 
5788 
5724 
5338 
5218 
8929 
3299 
0945 
6760 
8258 
5305 
Source: Arkin, Herbert; Table of 120,000 Random Decimal Digits, Bernard M. Baruch College, 1963. 