When constructing a confidence interval estimate, researchers must also know how the standard error of the mean is related to the standard deviation of the universe. In a simple random sample, that relationship is:
σx = σ / √n
where: σ x = Standard error of the mean
σ = Standard deviation of the universe
n= Number of observations in the sample.
This formula applies if less than 5 percent of the universe is included in the sample.
This relationship between the standard error of the mean σ x and the standard deviation of the universe( σ ) implies that when larger sample sizes (n) are used, the standard error of the mean will be a smaller value and, therefore, the mean of a given sample is likely to be closer to the universe mean.
Describing the Distribution of Sample Means:
Using their knowledge of (a) the relationship between the standard error of the mean and the universe standard deviation, and (b) the characteristics of the normal curve approximation of the distribution of sample means, researchers can describe the distribution of sample means about the universe mean. They can do so because:
1) The distribution of sample means is approximately normal, its mean is equal to the universe mean (M) and it has a standard deviation of
σ x = σ / √n
2) The ± Z ranges enable one to specify the percent of sample means lying within ± 1 standard error of the universe mean the percent of sample means lying within ± 2 standard errors of the universe mean, and so on.
Examples: To illustrate the application of these concepts, consider what can be said about the distribution of sample drawn as curve where the sample size is n=100. Assume that in the universe being sampled (drawn as curve A) the mean is $25,000 and the standard deviation is $40,000.
Knowledge of these three facts (sample size, universe mean and universe standard deviation) enables the researcher to describe (to an adequate approximation) the distribution of sample means shown as curve B. The mean of curve B will be $ 25,000, because the mean of the distribution of sample means is always equal to the universe mean. The standard error of the sample mean distribution is
σ x = σ / √ n, so that here σ x = $40,000 / √100 = $4,000
Because the distribution of sample means is approximately normal, this information on the universe mean ($25,000) and the standard error (σx=$4,000) provides a detailed description of curve B. For example, because 68.26 percent of sample means will fall within one standard of error of the universe mean, this implies that, for curve B, the interval.
Universe Mean ± 1 standard
= $25,000 ± (1) ($4,000)
= $21,000 — $29,000
Will include about 68.26 percent of sample means. Similarly, in curve B, about 95.44 percent of ample means will lie within the interval:
Universe Mean ± 2 standard Errors
= $25,000 ± 2 ($4,000)
= $ 17,000 — $ 33,000.
Virtually all (99.72 percent) of the sample under curve B will be located within the interval;
Universe Mean ± 3 Standard Errors
= $25,000 ± 3 ($4,000)
$13,000 — $ 37,000
The same technique may be applied to curve C, the difference being that n= 400 is used (instead of 100) in the calculation of σx. The calculations are left to the reader as an exercise.
In general, by using detailed tables (Appendix A), the probability that a sample mean will lie between any two numbers can be calculated. This assumes that the universe mean and standard deviation are known and that n is large. For most purposes n = 30 is sufficient, provided the universe is not excessively asymmetrical.