Statistical Methods for Psychology

The rate at which the sampling distribution of the mean approaches normal as nin-

creases is a function of the shape of the parent population. If the population is itself nor-

mal, the sampling distribution of the mean will be normal regardless of n. If the population

is symmetric but nonnormal, the sampling distribution of the mean will be nearly normal

even for small sample sizes, especially if the population is unimodal. If the population is

markedly skewed, sample sizes of 30 or more may be required before the means closely

approximate a normal distribution.

To illustrate the central limit theorem, suppose we have an infinitely large population

of random numbers evenly distributed between 0 and 100. This population will have what

is called a uniform (rectangular) distribution—every value between 0 and 100 will be

equally likely. The distribution of 50,000 observations drawn from this population is shown

in Figure 7.1. You can see that the distribution is very flat, as would be expected. For uni-

form distributions the mean (m) is known to be equal to one-half of the range (50), the stan-

dard deviation (s) is known to be equal the range divided by the square root of 12, which

in this case is 28.87, and the variance ( ) is thus 833.33.

Now suppose we drew 5000 samples of size 5 (n 5 5) from this population and plotted

the resulting sample means. Such sampling can be easily accomplished with a simple com-

puter program; the results of just such a procedure are presented in Figure 7.2a, with a nor-

mal distribution superimposed. It is apparent that the distribution of means, although not

exactly normal, is at least peaked in the center and trails off toward the extremes. (In fact

the superimposed normal distribution fits the data quite well.) The mean and standard

deviation of this distribution are shown, and they are extremely close to m550 and

. Any discrepancy between the actual values and those
predicted by the central limit theorem is attributable to rounding error and to the fact that
we did not draw an infinite number of samples.
Now suppose we repeated the entire procedure, only this time drawing 5000 samples
of 30 observations each. The results for these samples are plotted in Figure 7.2b. Here you

sX=s> 1 n=28.87> 15 =12.91

s^2

Section 7.1 Sampling Distribution of the Mean 181

Individual observations

1.05.09.013.017.021.025.029.033.037.041.045.049.053.057.061.065.069.073.077.081.085.089.093.097.0

Frequency

1200

1000

800

600

400

200

0

Figure 7.1 50,000 observations from a uniform distribution

uniform
(rectangular)
distribution