If X ~ Binomial(n, p) the Central Limit Theorem provides an approximation to the CDF FX. This approximation is given by
FX(k) ≈ Φ((k + 0.5 – np)/√ (npq)).
Here Φ is the CDF of a standard normal (Gaussian) random variable and q = 1-p. The central limit theorem approximation is studied in these notes. The Camp-Paulson approximation improves on the classical approximation by using a non-linear transformation of the argument k. This approximation uses
FX(k) ≈ Φ((c – μ)/σ)
c = (1-b)r1/3,
μ = 1 – a,
σ = √(br2/3 + a),
a = 1/(9n-9k),
b = 1/(9k+9), and
r = (k+1)(1-p)/(np–kp).
Johnson and Kotz prove that the error in the Camp-Paulson approximation is never more than 0.007/√ (npq). The Camp-Paulson approximation is often one or two orders of magnitude more accurate than the classical approximation arising directly from the Central Limit Theorem.
For more information, see “Some Suggestions for Teaching About Normal Approximations to Poisson and Binomial Distribution Functions” by Scott M. Lesch and Daniel R. Jeske, The American Statistician, August 2009, Vol 63, No 3.