Theorem 1: If x is a random variable with distribution B(n, p), then for sufficiently large n, the following random variable has a standard normal distribution:
Proof: Click here for a proof of Theorem 1, which requires knowledge of calculus.
Corollary 1: Provided n is large enough, N(μ,σ) is a good approximation for B(n, p) where μ = np and σ2 = np (1 – p).
Observation: The normal distribution is generally considered to be a pretty good approximation for the binomial distribution when np ≥ 5 and n(1 – p) ≥ 5. For values of p close to .5, the number 5 on the right side of these inequalities may be reduced somewhat, while for more extreme values of p (especially for p < .1 or p > .9) the value 5 may need to be increased.
Example 1: What is the normal distribution approximation for the binomial distribution where n = 20 and p = .25 (i.e. the binomial distribution displayed in Figure 1 of Binomial Distribution)?
As in Corollary 1, define the following parameters:
Since np = 5 ≥ 5 and n(1 – p) = 15 ≥ 5, based on Corollary 1 we can conclude that B(20,.25) ~ N(5,1.94).
We now show the graph of both pdf’s to see visibly how close these distributions are:
Figure 1 – Binomial vs. normal distribution