The Normal Dist. as an Empirical Fit to the Binomial Dist.

  The binomial distribution is a rather complicated function and the factorials are difficult to deal with so one might be tempted to seek a simpler function that approximates it. The binomial distribution is fairly symmetric about the mean value, μ, and a logarithmic plot reveals an approximate quadratic function so we might try to fit a function of the form,

This is a discrete probability function and its sum over all values of k is equal to 1 so A is actually a function of β too. We could try a search for β that minimizes the mean square error or find an approximate solution and try to improve on it. The second method gave the following fit.

The first plot compares the fit with the binomial distribution which is quite good in this example. The red points in the second plot show the deviation of the fit from the binomial distribution and the blue points give the same for the normal distribution formula using the  expected values <k> for μ and <k²> for λ. For most values of p the two error "curves" are nearly equal but for p near 1/2 the least squares fit has lower bounds on the error.

Supplemental (May 8): Recomputed the normal distribution error evaluating the erf function at k±0.5 to get:

The curvature of the binomial distribution near the peak and wings may be responsible for the deviations of the two approximation functions.