Handbook of Civil Engineering Calculations

The quantity JJL^ is an index of the diversity of the sample means. As the sample size in-

creases, the samples become less diverse.

Since a sample represents a combination of TV objects taken w at a time, the number of

samples that may be drawn is CN>n = N\/[n\(N- «)!].

SAMPLING DISTRIBUTION OF THE MEAN

The population consists of 5 objects having the numerical values 15, 18, 27, 36, and 54;
the sample size is 3. Find the mean and standard deviation of the sampling distribution of
the mean.

Calculation Procedure:

1. Compute the mean and variance of the population
   Mean IUL = (15 + 18 + 27 + 36 + 54)/5 = 30; variance o^2 = [(15 - 3O)^2 + (18 - 3O)^2 + (27 -
   3O)^2 + (36 - 3O)^2 + (54 - 30)^2 ]/5 = 198.


2. Compute the properties of the sampling distribution
   Apply Eq. 14 to find the mean of the sampling distribution of the mean, or /^ = 30. Ap-
   ply Eq. 15 to find the variance of the sampling distribution, or crj = 198(5 - 3)/(3 x 4) =


3. Then cry V33 = 5.74.


4. Compute the required properties without recourse to any
   set equations
   If the population is finite, the number of possible samples is finite. Since all samples have
   an equal likelihood of becoming the true sample, the sampling distribution of a statistic
   can be found by forming all possible samples and computing the statistic under consider-
   ation for each.
   Record all possible samples in the first column of Table 27; the number of these sam-
   ples is C 5 3 = 10. Now compute the mean X of each sample, record the results in the sec-

TABLE 27. Properties of Sampling Distribution of

the Mean

Sample Deviation d

Sample mean X = X-3Q d^2

15,18,27 20 -10 100

15,18,36 23 -7 49

15,18,54 29-1 1

15,27,36 26 -4 16

15,27,54 32 2 4

15,36,54 35 5 25

18,27,36 27 -3 9

18,27,54 33 3 9

18,36,54 36 6 36

27,36,54 39 9 81

Total 300 O 330