Simple approximation to normal distribution

Here’s a simple approximation to the normal distribution I just ran across. The density function is

f(x) = (1 + cos(x))/2π

over the interval (-π, π). The plot below graphs this density with a solid blue line. For comparison, the density of a normal distribution with the same variance is plotted with a dashed orange line.

PDF plots

The approximation is good enough to use for teaching. Students may benefit from doing an exercise twice, once with this approximation and then again with the normal distribution. Having an approximation they can integrate in closed form may help take some of the mystery out of the normal distribution.

The approximation may have practical uses. The agreement between the PDFs isn’t great. However, the agreement between the CDFs (which is more important) is surprisingly good. The maximum difference between the two CDFs is only 0.018. (The differences between the PDFs oscillate, and so their integrals, the CDFs, are closer together.)

I ran across this approximation here. It goes back to the 1961 paper “A cosine approximation to the normal distribution” by D. H. Raab and E. H. Green, Psychometrika, Volume 26, pages 447-450.

Update 1: See the paper referenced in the first comment. It gives a much more accurate approximation using a logistic function. The cosine approximation is a little simpler and may be better for teaching. However, the logistic approximation has infinite support. That could be an advantage since students might be distracted by the finite support of the cosine approximation.

The logistic approximation for the standard normal CDF is

F(x) = 1/(1 + exp(-0.07056 x3 – 1.5976 x))

and has a maximum error of 0.00014 at x = ± 3.16.

Update 2:

How might you use this approximation the other way around, approximating a cosine by a normal density? See Always invert.

Related posts:

Rolling dice for normal samples
Sums of uniform random variables

9 thoughts on “Simple approximation to normal distribution

  1. There’s a history of these … There’s a famous approximation by Polya which I first learned of in the Chmura Kraemer and Thiemann book How Many Subjects?:

    and an improvement in, I believe, a Master’s thesis by one Aludaat-Alodat:

  2. Another nice approximation is the piecewise-cubic
    M4 = M1 convolve M1 convolve M1 convolve M1
    where M1(x) is the step function: 1 for x in -.5 .. .5, else 0.
    Explicitly,

    def M4(x):
    return (
    np.fmax( x+2, 0 ) ** 3
    - 4 * np.fmax( x+1, 0 ) ** 3
    + 6 * np.fmax( x, 0 ) ** 3
    - 4 * np.fmax( x-1, 0 ) ** 3
    + np.fmax( x-2, 0 ) ** 3
    ) / 6

    This comes from a lovely paper by I.J.Schoenberg, “On equidistant cubic spline interpolation”, Bulletin AMS, 1971.

    cheers
    — denis

  3. Folklore has it that an early IBM standard normal distribution pseudo-random number generator was the sum of twelve [0,1] uniform pseudo-random numbers minus 6. A straight-forward application of the central limit theorem.

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