Central Limit Theorem

The central limit theorem concerns the distribution of a linear composite. Y is a linear composite of a set of variables (X1, X2, X3, etc.) if Y = a1 X1 + a2 X2 + a3 X3 + ... , where the a's are weights. For example, if Y equals 3X1 + 4X2, then a 1 equals 3 and a2 equals 4. The central limit theorem

Table 1. Distribution ofX

Value of X

Probability of X

0

.50

1

.50

I. Distribution of Y

Value of Y

Probability of Y

0

.001

1

.010

2

.044

3

.117

4

.205

5

.246

6

.205

7

.117

8

.044

9

.010

10

.001

states that the shape of the distribution of Y becomes more and more like the normal distribution as the number of variables included in the composite increases. Specifically, the central limit theorem states that Y is asymptotically normal as the number of composited variables approaches infinity. The central limit theorem is one of the principal reasons that psychologists and statisticians make regular use of the normal distribution.

Notice that the theorem does not require that the variables in the composite be normally distributed. Y is asymptotically normal even when the composited variables have very nonnormal distributions. Perhaps this is most easily illustrated by compositing a set of coin tosses. Imagine tossing a fair (unbiased) coin one time, recording 0 for a tail and 1 for a head. This experiment has two possible outcomes, each equally likely. If we call the experiment's outcomeX, it can be concluded that P (X = 0) = 0.5 and P (X = 1) = 0.5. The distribution ofX is given in Table 1.

Repeat this simple experiment 10 times, generating values forX1,X2,X3 . . .X10. Each oftheX's has the same distribution. It is possible to create a new variable that is a linear composite of the X's. Let Y = X1 + X2 + X3 + ... + X10. Y is the number of heads in 10 tosses of a fair coin, and the distribution of Y is given in Table 2. Notice that with only 10 variables in our composite, Y resembles the normal distribution; probabilities are highest in the middle of the distribution and gradually decrease for more extreme scores. If the coin were tossed 1,000 times and the outcomes were summed, the distribution of this sum would be almost indistinguishable from the normal distribution.

The central limit theorem frequently is introduced as a special case, to describe the distribution of the sample mean. The sample mean is a linear composite of the scores in the sample, with each score weighted by UN, where N is the sample size. IfN is large enough, the distribution of the sample mean will be normal, so the normal distribution can be used to build confidence interval estimates of the population mean and to test hypotheses concerning the sample mean. Researchers generally assume the sample mean has a normal distribution if.^ is at least 30, but how quickly the distribution assumes the normal shape depends on how normal theX's are. If theXscores are normally distributed, their mean always will be normally distributed. If the X scores are very nonnormal, N may have to be larger than 30 for the distribution to be normal.

The central limit theorem also can explain why many physical measurements are normally distributed. Human heights and weights are determined by many factors, probably including hundreds of genes and thousands of variables related to nutritional and psychological history. Heights or weights can be thought of as composite of thousands of variables, so they should be normally distributed. Many psychological traits, such as intelligence, also are normally distributed, probably because they are influenced by thousands of genes and prenatal and postnatal events. Deviations from normality suggest the heavy influence of one event that overrides the linear composite. For example, people who are so extremely short that their presence is not consistent with the normal distribution may have heights determined by a pituitary problem. Their heights are not influenced by the genes and events that under other circumstances would have made them taller. Similarly, people with extremely low intelligence may have a rare genetic defect or may have been subject to some trauma that damaged the central nervous system.

Mary J. Allen

California State University

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