Download Introduction to Sampling Theory

AN INTRODUCTION TO
SAMPLING THEORY
15
Introduction and Terminology
Statistical analysis may be performed on data from either a population or a sample. A population is the entire
universe or entire set of observations, while a sample represents a subset of the population. The population may
change from one problem to the next. For example, the population may be all of the houses in Vancouver, or
the population may be all of the houses in a particular neighbourhood in Vancouver, or the population may be
all of the houses which sold in that neighbourhood in Vancouver in a given year. A sample of five houses might
be drawn from each of these three populations. In the examples presented thus far in this text, it has been
implicitly assumed that the data constituted a population rather than a sample.
While the discussions in Chapter 12 defined the mean and standard deviation for the population, these measures
also exist for samples. These measures, when calculated for samples, are called sample statistics. Sample
statistics have distributions (and therefore means and standard deviations) and are related to population statistics;
however, the derivation of their distributions is beyond the scope of this text.
When the observations in a population are being analyzed, full information is available because all of the relevant
observations are present. However, there are many cases when only a sample from a population is available
for analysis. This may occur for several reasons.
(1) It may be difficult and costly to define or find every observation in the population. For example, if the
population consists of every housing unit in a large city, it could be difficult to find and list each item
in the population.
(2) Once the observations are defined, it may be expensive to obtain the necessary data for every
observation in the population. For example, it would be expensive to determine the number of square
feet in every housing unit in a large city.
(3) It may not be feasible or reasonable to obtain the necessary data for every observation. For example,
if the problem is to find out how long light bulbs last, it would not be reasonable to let every one burn
until it burned out since no useful light bulbs would exist after all the data had been collected.
For these reasons, it is often necessary to deal with samples rather than populations. The focus of this chapter
will be on sampling theory, and in particular, methods of drawing samples. Problems associated with samples
will also be discussed.
Sampling and Probability
Samples are used in the circumstances described above as a method of determining information about the
population. It is possible that the observations in the sample will not be "representative" of the observations in
the population. A sample is representative of the population if the characteristics of the observations in the
population are reflected in roughly the same proportions in the sample. For example, a sample of ten housing
units in a large city might all have values over $500,000. Such a sample will not be representative of the
15.1
Chapter 15
population of all housing units in the city if only 1% of the population have values over $500,000. Also, it is
often claimed that the houses sold through a multiple listing service are not representative of the population of
all houses sold in a municipality. Below, several methods will be discussed for obtaining samples, some of
which are less likely to produce non-representation errors than others.
While sampling involves inferences about the population, given information about a sample, probability theory
involves making inferences about samples, given information about the population. For example, one might ask
for the probability that a sample of housing units would contain units with a certain value. Forming probabilities
may be relatively simple — picking a head or tail when a fair coin is flipped — or relatively difficult — assigning
a probability to the potential success or failure of a new shopping centre. A more detailed discussion of
probability theory is beyond the scope of this text; however, the reader should be aware of the relationship
between sampling theory and probability theory as discussed above.
Methods of Sampling
As noted above, a sample may or may not be representative of the population from which it is chosen. In this
section, several methods of drawing samples will be presented. If samples are chosen randomly (to be discussed
below), then sampling theory may be applied to make certain estimates of the potential error due to the sampling
process. If samples are not chosen randomly, it is much more difficult to make such estimates.
Random sampling requires that every item in the population have an equal and independent chance of being
included in the sample. For this to occur, each item in the population must be identified, and there must be a
selection process that ensures that each item in the population has an equal and independent chance of being
chosen. This technique is also called probability sampling.
To draw a random sample of size "n", it is first necessary to assign a number to each observation in the
population, beginning with some number (usually one) and numbering each observation in the population
consecutively, continuing as far as is necessary, say m. It is then necessary to choose a set of n random numbers
between 1 and m, inclusive. One way to do this is with a random number table, such as the one shown as Table
1 at the end of this chapter. There are a variety of ways to construct a random number table. Conceptually,
the easiest way is to place ten balls, each of which is labelled with a single digit between zero and nine inclusive,
into an urn. One ball is then chosen at random, its value recorded in the first row and column in the table, and
then placed back in the urn. Another ball is then chosen with its value being recorded in the first row of the
second column. After it is replaced, the process is repeated until the table is fully populated. The use of this
type of table is shown below.
Illustration 15.1
Assume there are 50 items in a population and a random sample of size 10 is desired. Table 1 may be used to
draw the sample. Any column or set of columns may be chosen and then searched for the first ten entries less
than or equal to 50. The fifth and sixth digit columns may be treated as a column of double digits. Reading
down the newly created column, the numbers 73, 20, 26, 90, 79, 57, 01, 97, etc., are encountered. Because
there are only 50 observations in the population, a random number must be less than or equal to 50 to be usable.
Thus, the first ten distinct random numbers encountered in that column are: 20, 26, 01, 33, 50, 29, 46, 11, 43,
and 09. The data items corresponding to these ten numbers constitute a random sample of size 10 drawn from
a population of size 50. If the first two single digit columns had been used, the random sample would consist
of items 10, 37, 08, 12, 31, 11, 9, 44, 12, and 15. This is obviously a different sample, but it is also random.
15.2
Introduction to Sampling Theory
Illustration 15.2
Suppose a population has 100 items and a random sample of five items is desired. The 100 items could be
numbered from 101 to 200. The sample can be chosen by choosing any three single-digit columns and searching
for random numbers between 101 and 200. If the first, third, and fifth single-digit columns are used, the first
ten random numbers are 107, 352, 042, 909, 187, 605, 300, 829, 653, and 776. Most of these are too large.
The first five usable random numbers are 107, 187, 185, 150, and 144. The items corresponding to these five
numbers would constitute a random sample of size five. Clearly, the use of different columns from Table 1 is
likely to yield a different random sample.
Example 15.1
Statement of Problem:
From a population of 25, draw a random sample of size 5 using the seventh and eighth single digit columns
of the random number table.
Solution:
The sample consists of the five observations corresponding to the first five distinct random numbers less
than or equal to 25 which are 25, 08, 21, 02, and 05.
If a random number table is not available, the numbers from a telephone directory could be used. It has been
shown that of the seven digits in a telephone number, the first four and the last one are not randomly distributed;
however, the fifth and sixth digits are virtually randomly distributed. Thus, these two columns may be used to
generate random numbers. If three or more digit numbers are required, the numbers could be formed by taking
successive vertical combinations of digits.
The procedure described above using a random number table allows the drawing of a truly random sample. The
use of this procedure increases the probability that the sample will be representative of the population; however,
it does not guarantee that it will be because the probability that each item is chosen is independent of and equal
to the probability that any other item is chosen.
Thus, the procedure does not prohibit the choosing of a large house despite the fact that other large houses have
already been chosen. Because of this possibility, a stratified random sample is sometimes drawn. This type of
sample might be used where clear, discernible groups or strata exist and when it is desirable to obtain a sample
which reflects the stratified nature of the population. For example, in examining the distribution of housing
values in a large housing market, it might be useful to define three strata consisting of large, medium and small
houses. If the proportions in the population for these types are 10%, 60%, and 30% respectively, it is possible
that large houses would be under-represented, even in a random sample. In this case, a stratified random sample
could be generated by choosing a random sample of the desired size from each of the strata. If the population
has 100,000 housing units and a stratified random sample of 1,000 is desired, the procedure would be to
combine a random sample of size 100 from the 10,000 large houses, a random sample of size 600 from the
60,000 medium houses, and a random sample of size 300 from the 30,000 small houses.
While a random sample has the highest probability of being representative, it is sometimes impractical. Its
application requires that all of the items in the population must be delineated and numbered. This can be a
problem if the population is large, as is the population of all housing units in a large metropolitan area. In such
cases, other sampling techniques are used.
15.3
Chapter 15
A systematic sample is generated by choosing every nth item in a list of the population until the required sample
size has been reached. A systematic sample of houses in a metropolitan area could be obtained by choosing
every 100th house in a list of all houses obtained from an assessment authority. Although this technique is easy
to use, the potential for a non-random sample exists. A sample chosen in this fashion can be random only if:
(1) the first item is chosen randomly;
(2) the value of n is chosen randomly; and
(3) the listing of items does not reflect a regular pattern (such as alphabetical ordering, or ascending order).
This is referred to as a lack of order bias.
A problem could exist if every twentieth house is chosen from a list of houses arranged by neighbourhood
because the houses chosen are unlikely to be representative of the population. Similarly, a problem could arise
if a production line inspector chooses every 100th item off the line, and one of several machines performing the
same function of the line is deficient. The number of defective items found could be very low if the machine
does not produce any items that are selected by the inspector, or very high if the inspector over-samples the
output of the defective machine. In both cases the sample would likely be non-representative. Although the
samples chosen in this fashion are often usable, they are not true random samples.
A quota sample is similar to a stratified sample in that it takes account of the distribution of characteristics in
the population. The difference between a quota sample and a stratified sample is that the former need not be
chosen so that the characteristics in the sample are in the same proportion as in the population. Using the
information from the discussion on stratified samples, a quota sample could be chosen so that the number of
large, medium and small houses in the sample are equal despite the fact that they occur with different relative
frequencies in the population. Presumably the items in the sample would be chosen on a random basis. A quota
sample chosen in this manner is clearly not representative of the population, and thus, generalizing about the
population may be difficult.
A judgment sample is one in which items are chosen from the population for inclusion in the sample based on
the analyst's judgment. The goal is to draw a representative sample based on the analyst's information and
knowledge of the population. A real estate agent might be asked to provide a sample of ten housing units which
are representative of the population of housing units in the city. The agent's knowledge and information of the
market would facilitate the choosing of such a sample. While such a sample may be representative, it will not
be a true random sample, and thus care must be taken in drawing inferences from it.
Sometimes it is useful to draw a convenience sample where the sole criterion for inclusion in the sample is that
it is convenient for the sampler to include the observation. Such a sample would be useful in the early stages
of research, for example, in the design of a survey instrument. Similarly, it may be useful to sample "the man
on the street" to obtain opinions. Such a sampling technique is useful when used in an exploratory fashion, but
it is subject to substantial bias.
Clusters, or area samples, are based on a division of the population into groups or clusters. If it is necessary
to sample households regarding their retail buying behaviour, then one method would be to take a random
sample of all households in the city. An alternative method would be to take a random sample of census tracts
within the city. Within each chosen census tract, one could then choose a random sample of enumeration areas.
Within each enumeration area, a set of randomly chosen households could then be surveyed. This technique
does not require the delineation of every household in the city, and thus, is often less costly than a truly random
sample.
In this section, seven sampling methods have been presented. In general, it is desirable to use random sampling
techniques as they increase the probability that the sample will be representative of the population and allow
statistical inferences to be made soundly. Non-random techniques may be used when constraints such as data
availability and cost exist, but they are less reliable.
15.4
Introduction to Sampling Theory
Illustration 15.3
When an appraiser selects comparables to use to determine the value of a housing unit, a judgment sample is
being taken as the appraiser uses his judgment to select comparables from the population of all housing units.
Illustration 15.4
Statistics Canada collects and compiles data for several types of geographical areas such as enumeration areas,
census tracts and census metropolitan areas. The housing data that Statistics Canada collects comes from a 100%
sample; that is, they collect information from every housing unit. Other data, such as labour force participation
data, is obtained from a 33.3% sample; that is, only one in every three households are sampled.
Illustration 15.5
Canada Mortgage and Housing Corporation collects extensive monthly data on housing starts and completions
in Canada. An example of this data is provided in Example 15.2. This data represents the population of starts
and completions rather than a sample. Because of this, the analyst can be very confident that the data are
accurate without being concerned with the possible non-representativeness of a sample.
Example 15.2
Statement of Problem:
Do houses listed with a multiple listing service constitute a random sample of all housing units? Why or
why not?
Solution:
No, because MLS generally over-represents lower priced houses and under-represents higher priced houses.
Example 15.3
Statement of Problem:
If the first ten drivers across a bridge each morning are surveyed, what type of sample do they constitute?
Solution:
If more information were provided, it might be possible to determine if the drivers represented a quota
sample. In the absence of more information, the sample would be called a convenience sample.
It is important to restate the fact that a sample will not necessarily be representative just because it is random.
If we know nothing about the characteristics of the population, then choosing a random sample increases the
probability that it is representative but does not guarantee that it will be. Since in general we do not know the
distribution of characteristics in the population we actually do not even know if the sample is representative.
15.5
Chapter 15
Sample Bias
Sample bias, or errors, may result in any sampling problem. If random samples are used, then it is possible to
talk about the probable size of the error. If a random sample is not used, it is much more difficult to make such
a statement. Bias can also result from several other aspects of the sampling process, such as:
(1) A flaw in the sample design. If a stratified or quota sample is used, it is necessary to have information on
the distribution of certain characteristics of the population. If that information is incorrect, then the sample
may be generated incorrectly. As a result, bias can occur.
(2) Non-response error. It is often the case that respondents do not respond to all of the questions that might
be asked in a survey or questionnaire. Errors that are caused by differences in the characteristics of
respondents and non-respondents are referred to as non-response error. The likelihood that non-response
error will be a problem increases as the response rate decreases because respondents and non-respondents
seldom have the same characteristics.
(3) Interviewer bias. This type of bias occurs when the interviewee's response is affected by the interviewer.
This may occur because of the way the interviewer asks the question, probes into a response, or gives
intentional or unintentional cues to the respondent.
These types of errors can pose substantial problems. While increasing the sample size can help with the
sampling error in a perfectly random sample, the types of errors described above cannot be minimized by
increasing the sample size. Rather, careful planning of the sampling procedure can help minimize the impact
of these errors.
Statistical Reliability
The underlying concern in this chapter has been the degree to which the characteristics of the population are
reflected in a sample drawn from the population. If the sample is not randomly selected, the probability that the
sample is not representative will increase. Similarly, if sampling bias is present, there will be problems in
making statistical inferences about the population given information about the sample.
For example, suppose that there is a need to obtain an estimate of the mean value for all houses in a large
housing market. One method of determining this value would be to obtain the value of every house in the
market and then calculate the mean. Alternatively, a random sample of some size could be generated and then
the mean value could be calculated for the sample. This sample mean is an estimate of the population mean.
The more representative the sample is of the population, the better an estimate the sample mean will be of the
population mean. On the other hand, if the sample is not very representative of the population, then the sample
mean may not be a good estimate of the population mean. Thus, the sample mean may or may not be a very
good estimate of the population mean.
To deal with this problem, we might draw a second random sample of the same size so as to obtain a second
estimate of the population mean. If the second sample mean is close to the first sample mean, that would
increase the confidence in either as an estimate of the population mean. If the two are quite different, then it
might be concluded that one or both of the samples were not very representative of the population. Thus, a third
random sample might be drawn. This process could be repeated until the sample means cluster enough so that
the analyst can be confident about the population mean. An illustration will help clarify the process.
15.6
Introduction to Sampling Theory
Illustration 15.6
Suppose the population consists of four houses (labelled A, B, C, D) with the following values:
A.
B.
C.
D.
$120,000
$130,000
$170,000
$180,000
and suppose that samples of size two are drawn. Before proceeding, it should be noted that this example is
simplified considerably. First, the population would normally be much larger, the values would not all be
known easily, and the population mean would not be so easily calculated.
Second, the sample size would normally be less than 5% of the population rather than 50%. Third, samples
would normally be drawn randomly, while in this example, all possible samples will be discussed.
Given these simplifications, the possible samples of size two and their respective sample means are:
A,B:
A,C:
A,D:
B,C:
B,D:
C,D:
$125,000 = (120,000 + 130,000)/2
$145,000 = (120,000 + 170,000)/2
$150,000 = (120,000 + 180,000)/2
$150,000 = (130,000 + 170,000)/2
$155,000 = (130,000 + 180,000)/2
$175,000 = (170,000 + 180,000)/2
It is clear that some samples yield a sample mean that is close to the population mean ($150,000), while some
do not yield good estimates of the population mean.
This illustration shows that the sample mean (denoted x ) has several possible values and may be treated like any
variable which takes on several values. Thus, the sample mean has a distribution, and its distribution has a mean
and a standard deviation. The standard deviation of the distribution of sample means is referred to as the
standard error of the mean. This standard error of the mean is a measure of the variation, or sampling error,
that occurs due to chance. The larger the standard error of the mean, the poorer the sample mean is as an
estimator of the population mean. The standard error of the mean is a measure of the statistical reliability of
the sample.
Although it is beyond the scope of this text, it can be shown that under certain circumstances, the standard error
of the mean can be given by:
Fx =
F
n
(Equation 15.1)
where
Fx = the standard error of the mean
F = the standard deviation of the population
n = the sample size
Some illustrations of this formula follow.
15.7
Chapter 15
Illustration 15.7
Using the data from Illustration 15.6, the standard deviation of the population is $25,495.10. Thus, the standard
error of the mean is:
Fx =
F
n
=
25,495.10
2
=
25495.10
= $18,027.76
1.41421356
Illustration 15.8
Using the data from Illustration 15.6, the process can be repeated for samples of size 3 given the same
peculiarities of that illustration. The possible samples and their sample means are
A,B,C:
A,B,D:
A,C,D:
B,C,D:
Fx =
$140,000 = (120,000 + 130,000 + 170,000)/3
$143,333 = (120,000 + 130,000 + 180,000)/3
$156,667 = (120,000 + 170,000 + 180,000)/3
$160,000 = (130,000 + 170,000 + 180,000)/3
F
n
=
25,495.10
3
=
25,495.10
= $14,719.60
1.7320508
Notice that the standard error of the mean decreased as the sample size increased from 2 to 3. That is, there
is less variation in the sample mean with three, as opposed to two, items in the sample.
Several statements about statistical reliability can be made given that the samples are chosen randomly. These
statements follow from Equation 15.1.
(1) Statistical reliability is independent of the population size as long as the sample constitutes less than 5%
of the population because the population size does not appear in Equation 15.1. Notice that the 5% rule
is violated by the data in Illustrations 15.6, 15.7, and 15.8.
(2) Statistical reliability increases in proportion to the square root of the sample size because only the square
root of the sample size appears in Equation 15.1. To reduce the sampling error by one-half, it is
necessary to increase the sample size by a factor of four. Thus, a sample of 100 is only twice as
reliable as a sample of 25 when both are drawn randomly from the same population.
(3) Statistical reliability of the sample is directly dependent on the amount of variation (the standard
deviation) in the population. This is so because the population standard deviation appears in Equation 1.
Statement 1 above may seem counter-intuitive because it states that as long as the sample size is small relative
to the population, then the statistical reliability of the sample is unaffected by the population size.
15.8
Introduction to Sampling Theory
Illustration 15.9
If the standard deviation of the population is 1000, what sample size is necessary so that the standard error of
the mean is 100?
From Equation 15.1, it follows that:
n × Fx = F and thus, n =
F
F2
and finally, n =
2
Fx
Fx
Now by substitution,
n=
10002
2
100
=
1,000,000
= 100
10,000
Example 15.4
Statement of Problem:
If the sample size is 49 and the standard error of the mean is 10, what is the standard deviation of the
population?
Solution:
F = 70
Example 15.5
Statement of Problem:
What sample size is needed if the standard deviation of a population is 225 and the desired standard error
of the mean is 45?
Solution:
n = 25
Summary
This chapter represents a shift in focus from populations to samples. Although it is desirable to work with
populations, it is often necessary to use samples instead. While many sampling techniques are available, it was
argued that the best sampling technique to use is random sampling. Whenever a sample is employed, it is
possible that it will not be representative of the population. Non-random samples and sampling bias can result
in such problems. A measure of statistical reliability was introduced to provide an estimate of how reliable a
random sample is.
15.9
Chapter 15
Table 1
Table of Random Numbers*
10 09 73 25 33
37 54 20 48 05
08 42 26 89 53
99 01 90 25 29
12 80 79 99 70
66 06 57 47 17
31 06 01 08 05
85 26 97 76 02
63 57 33 21 35
73 79 64 57 53
98 52 01 77 67
11 80 50 54 31
83 45 29 96 34
88 68 54 02 00
99 59 46 73 48
65 48 11 76 74
80 12 43 56 35
74 35 09 98 17
69 91 62 68 03
09 89 32 05 05
91 49 91 45 23
80 33 69 45 98
44 10 48 19 49
12 55 07 37 42
63 60 64 93 29
61 19 69 04 46
15 47 44 52 66
94 55 72 85 73
42 48 11 62 13
23 52 37 83 17
04 49 35 24 94
00 54 99 76 65
35 96 31 53 07
59 80 80 83 91
46 05 88 52 36
32 17 90 05 97
69 23 46 14 06
19 56 54 14 30
45 15 51 49 38
94 86 43 19 94
98 08 62 48 26
33 18 51 62 32
80 95 10 04 06
79 75 24 91 40
18 63 33 25 37
74 02 94 39 02
54 17 84 56 11
11 66 44 98 83
48 32 47 79 28
69 07 49 41 38
76 52 01 35 86
64 89 47 42 96
19 64 50 93 03
09 37 67 07 15
80 15 73 61 47
34 07 27 68 50
45 57 18 24 06
02 05 16 56 92
05 32 54 70 48
03 52 96 47 78
14 90 56 86 07
29 80 82 77 32
06 28 89 80 83
86 50 75 84 01
87 51 76 49 69
17 46 85 09 50
17 72 70 80 15
77 40 27 72 14
66 25 22 91 48
14 22 56 85 14
68 47 92 76 86
26 94 03 68 58
85 15 74 79 54
11 10 00 20 40
16 50 53 44 84
26 45 74 77 74
95 27 07 99 53
67 89 75 43 87
97 34 40 87 21
73 20 88 98 37
75 24 63 38 24
64 05 18 81 59
26 89 80 93 54
45 42 72 68 42
01 39 09 22 86
87 37 92 52 41
20 11 74 52 04
01 75 87 56 79
19 47 60 72 46
36 16 81 08 51
45 24 02 84 04
41 94 15 09 49
96 38 27 07 74
71 96 12 82 96
98 14 50 65 71
77 55 73 22 70
80 99 33 71 43
52 07 98 48 27
31 24 96 47 10
87 63 79 19 76
34 67 35 48 76
24 80 52 40 37
23 20 90 25 60
38 31 13 11 65
64 03 23 66 53
36 69 73 61 70
35 30 34 26 14
68 66 57 48 18
90 55 35 75 48
35 80 83 42 82
22 10 94 05 58
50 72 56 82 48
13 74 67 00 78
36 76 66 79 51
91 82 60 89 28
58 04 77 69 74
45 31 82 23 74
43 23 60 02 10
36 93 68 72 03
46 42 75 67 88
46 16 28 35 54
70 29 73 41 35
32 97 92 65 75
12 86 07 46 97
40 21 95 25 63
51 92 43 37 29
59 36 78 38 48
54 62 24 44 31
16 86 84 87 67
68 93 59 14 16
45 86 25 10 25
96 11 96 38 96
33 35 13 54 62
83 60 94 97 00
77 28 14 40 77
05 56 70 70 07
15 95 66 00 00
40 41 92 15 85
43 66 79 45 43
34 88 88 15 53
44 99 90 88 96
89 43 54 85 81
20 15 12 33 87
69 86 10 25 91
31 01 02 46 74
97 79 01 71 19
05 33 51 29 69
59 38 17 15 39
02 29 53 68 70
35 58 40 44 01
*
80 95 90 91 17
20 63 61 04 02
15 95 33 47 64
88 67 67 43 97
98 95 11 68 77
65 81 33 98 85
86 79 90 74 39
73 05 38 52 47
28 46 82 87 09
60 93 52 03 44
60 97 09 34 33
29 40 52 42 01
18 47 54 06 10
90 36 47 04 93
93 78 56 13 68
73 03 95 71 86
21 11 57 82 53
45 52 16 42 37
76 62 11 39 90
96 29 77 88 22
94 75 08 99 23
53 14 03 33 40
57 60 04 08 81
96 64 48 94 39
43 65 17 70 82
65 39 45 95 93
82 39 61 01 18
91 19 04 25 92
03 07 11 20 59
26 25 22 96 63
61 96 27 93 35
54 69 28 23 91
77 97 45 00 24
13 02 12 48 92
93 91 08 36 47
86 74 31 71 57
18 74 39 24 23
66 67 43 68 06
59 04 79 00 33
01 54 03 54 56
39 09 47 34 07
88 69 54 19 94
25 01 62 52 98
74 85 22 05 39
05 45 56 14 27
52 52 75 80 21
56 12 71 92 55
09 97 33 34 40
32 30 75 75 46
10 51 82 16 15
39 29 27 49 45
00 82 29 16 65
35 08 03 36 06
04 43 62 76 59
12 17 17 68 33
11 19 92 91 70
23 40 30 97 32
18 62 38 85 79
83 49 12 56 24
35 27 38 84 35
50 50 07 39 98
52 77 56 78 51
68 71 17 78 17
29 60 91 10 62
23 47 83 41 13
40 21 81 65 44
14 38 55 37 63
96 28 60 26 55
96 40 05 04 18
54 38 21 45 98
37 08 92 00 48
42 05 08 23 41
22 22 20 64 13
28 70 72 58 15
07 20 73 17 90
42 58 06 05 27
33 21 15 94 66
92 92 74 59 73
25 70 14 66 70
05 52 28 25 62
65 33 71 24 72
23 28 72 95 29
90 10 33 93 33
78 56 52 01 00
70 61 74 29 41
85 39 41 18 38
97 11 89 63 38
84 96 28 52 07
20 82 66 95 41
05 01 45 11 76
35 44 13 18 80
37 54 87 30 43
94 62 46 11 71
00 38 75 95 79
77 93 89 19 36
80 81 45 17 48
36 04 09 03 24
88 46 12 33 56
15 02 00 99 94
01 84 87 69 38
Source: W.J. Dixon and F.J. Massey Jr., Introduction to Statistical Analysis, 2nd Edition, McGraw-Hill Book Company, Inc., 1957, p. 366.
15.10