# Illustration of the central limit theorem

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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.

A more concrete illustration, in which most of the arithmetic can be done more-or-less instantly by hand, is at concrete illustration of the central limit theorem. There is also a free full-featured interactive simulation available which allows to set up various distributions and adjust the sampling parameters (see "external links" at the bottom of this page).

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.

The convolutions were computed via the discrete Fourier transform.
A list of values *y* = *f*(*x*_{0} + *k* Δ*x*) was constructed, where *f* is the original density function, and Δ*x* is approximately equal to 0.002, and *k* is equal to 0 through 1000.
The discrete Fourier transform *Y* of *y* was computed.
Then the convolution of *f* with itself is proportional to the inverse
discrete Fourier transform of the pointwise product of *Y* with itself.

We start with a probability density function. This function, although discontinuous, is far from the most pathological example that could be created. The mean of this distribution is 0 and its standard deviation is 1.

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. The density shown in the figure at right has been rescaled by √2 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.

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. The density shown in the figure at right has been rescaled by √3 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.

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. The density shown in the figure at right has been rescaled by √4 = 2 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.