Central Limit Theorem

The central limit theorem states that, under conditions of repeated sampling from a population, the sample means of random measurements tend to possess an approximately normal distribution. This is true for population distributions that are normal and decidedly not normal.

The central limit theorem is important for two reasons. The first can be understood by examining the fuzzy central limit theorem, which avers that whenever an attribute is influenced by a large number of

relatively independently occurring variables, the attribute will be normally distributed. Such attributes are ordinarily complex variables that require considerable time in their development. For instance, a multitude of relatively independently occurring variables influence adult reading achievement, each of which may be influential in large or small ways:

• 1.
  Quality of teachers in the early grades
• 2.
  Variety of reading matter available (books, magazines, newspapers)
• 3.
  Parental encouragement to read
• 4.
  Genetic endowment
• 5.
  Diet during developmental years
• 6.
  Socioeconomic level of the child's family
• 7.
  Number of competing activities available (e.g., arcade games, sports, television)
• 8.
  Interests and activities of friends
• 9.
  Initial success (or failure) in reading activities
• 10.
  Physical athleticism of the child

It is easy to imagine many other such variables. To the extent that the variables produce small, relatively random effects on adult reading achievement over time, the distribution of adult reading achievement will be normally distributed. Because the variables that influence such attributes produce unrelated effects, one's position in the distribution of scores in the population is largely a matter of luck. Some individuals are very lucky (e.g., they may have parents who encourage reading, access to reading materials, excellent early teachers) and enjoy high adult reading achievement. Some are very unlucky (e.g., they may have impoverished parents who do not support reading, poor diet, frequent illnesses) and remain illiterate. But about two thirds are neither very lucky nor very unlucky and become average adult readers. The fuzzy central limit theorem then, provides an explanation for the relatively common occurrence of normal distributions in nature.

The second reason the central limit theorem is important concerns its use in statistical inference. Because in a normal distribution we know the [Page 128]percentage of cases falling between any two points along the baseline and the percentage of cases above and below any single point along the baseline, many useful inferences can be made. Consider an investigation comparing two treatment approaches designed to enhance math achievement. Following the treatment, the dependent variable, a test of math achievement, is administered. By comparing the two treatment group means, the researchers determine the probability that the difference between the means could have occurred by chance if the two treatment groups were drawn from the same population distribution. If the probability were, say, 0.0094, the investigators could have a high degree of confidence in rejecting the null hypothesis of no difference between the treatments. If the central limit theorem did not commonly apply in nature, such inferences could not be made.