Why is the normal distribution so very normal?
The reason for the frequent appearance of the normal distribution among empirical variables and theoretical probability distributions is much simpler than you might imagine. The tendency of empirical events or theoretical entities to form a normal distribution is somewhat analogous to the tendency of water to run down a hill—it is simply the easiest and most natural way of going. In order to have water run down a hill, all we need is water and a hill. In order to have empirical events or theoretical entities form a normal distribution, all we need is the summation—the combined additive result—of a multiplicity of random coincidences. This simple but very important principle is embodied on the formal side of probability theory by what is known as the central limit theorem, which demonstrates mathematically that the sums of a multiplicity of random variates will tend to produce a normal distribution.
Thus, for any particular toss of a coin the outcome will include either 1 head or zero heads. Essentially, it is a random variable with two possible values, 1 and 0, for which the probabilities are p=.5 and q=.5, respectively. So if you toss the coin N times, counting up the total number of heads that occur, you are in effect taking the sum of N random variates. Repeat the N-toss operation a large number of times, and the result will be an approximately normal distribution of such sums—providing that the size of N is equal to or greater than 10 (see main text). Similarly, for any particular patient the outcome will include either 1 recovery or zero recoveries, with probabilities (in the absence of effective treatment) of p=.4 and q=.6, respectively. So when you count up the number of recoveries in a sample of N patients, you are again taking the sum of N random variates. Perform the same count on a large number of such samples, and the result will once again be an approximately normal distribution—providing that the size of N is equal to or greater than 13 (see main text).
The same reasoning applies to cases, such as the distributions of scores on a standardized IQ test, that are not essentially binomial in nature. The tendency for IQ scores to be normally distributed reflects the fact that each individual score is the result of a large number of randomly coincident causal factors—among them, one set of factors pertaining to the person's original hereditary endowment, another pertaining to the accumulated effects of environment and experience, and still another relating to the particular circumstances that prevail at the time the test is taken.
An effective way to gain a good intuitive grasp of how normal distributions come about is to generate one with your own hands, using only the raw materials of summed multiple chance coincidences. The apparatus and procedure for this experiment are very simple. First, at the bottom of a sheet of paper lay out the numbers 4 through 24, grouped in the following intervals:
| 4, 5, 6
| 7, 8, 9
| 10, 11, 12
| 13, 14, 15
| 16, 17, 18
| 19, 20, 21
| 22, 23, 24
|
Then take four dice, toss them, record the sum of the faces that come up, and place an X above the corresponding interval in the scale at the bottom of your sheet of paper. (As each die has six faces, numbered 1 through 6, any sum of four dice will fall between 4 and 24, inclusive.) And then, repeat this same operation again and again, several hundred times over, on each occasion placing an X above the corresponding interval on the scale. The longer you keep at it, the greater will be the resemblance that you find emerging between your stacks of X's and the familiar bell-shaped outlines of the normal distribution.