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Illustration of the central limit theorem

Here is an illustration of the central limit theorem. A probability density function is shown in the first figure. Then the densities of the sums of two, three, and four independent variables, each having the original density, are shown in the later figures.

Although the original density is far from normal, the density of the sum of just a few variables with that density is much smoother and has some of the qualitative features of the normal density.

The densities of the sums of two, three, and four terms were constructed as the convolution of the original density with itself. As the original density is a piecewise polynomial (of degree 0 and 1), the convolutions are also piecewise polynomials, of increasing degree. Thus the convolution of the original density may be considered a means of constructing a piecewise polynomial approximation to the normal density.

(An excellent companion article would be an example in which the central limit theorem does not apply, or at least shows much slower convergence.)

A probability density function

We start with a probability density function. This function, although discontinuous, is far from the most pathological example that could be created.

This density has mean 0 and standard deviation 1.

Density of a sum of two variables

Next we compute the density of the sum of two independent variables, each having the above density. The density of the sum is the convolution of the above density with itself.

The sum of two variables has mean 0 but it has standard deviation √2. The density shown in the figure at right has been rescaled so that its standard deviation is 1.

This density is already smoother than the original. There are obvious lumps, which correspond to the intervals on which the original density was defined.

Density of a sum of three variables

We then compute the density of the sum of three independent variables, each having the above density. The density of the sum is the convolution of the first density with the second.

The sum of three variables has mean 0 but it has standard deviation √3. The density shown in the figure at right has been rescaled so that its standard deviation is 1.

This density is even smoother than the preceding one. The lumps can hardly be detected in this figure.

Density of a sum of four variables

Finally, we compute the density of the sum of four independent variables, each having the above density. The density of the sum is the convolution of the first density with the third.

The sum of four variables has mean 0 but it has standard deviation 2 = √4. The density shown in the figure at right has been rescaled so that its standard deviation is 1.

This density appears qualitatively very similar to a normal density. Any lumps cannot be distinguished by the eye.