Normal Approximation to the Binomial Distribution

The random variable X that is binomially distributed with parameters p and n has the following probability mass function:

The mean and variance of X are

E(X) = np
Var(X) = np(1-p)

However, if the value of n is large, the z scores of X will have a probability distribution that approximates the standard normal distribution. That is,

The normal approximation to the binomial distribution is very accurate when n is large. When n is small, it still provides a fairly good estimate if p is close to 0.5. A useful guide is provided by calculating the values of np and n(1-p); if both values are greater than 5, the normal approximation to the binomial distribution will provide a good estimate (and forego the need of calculating tedious binomial probability formulas).

EX. The random variable X, the number of heads in 100 tosses of a coin, is binomially distributed with n = 100 and p = 0.5. The z score of X is

Since np = np(1-p) = (100)(0.5) = 50 > 5, the normal approximation to the binomial distribution should provide a good estimate. The probability that X will have a value between 48 and 58 is calculated as follows:

P(48 < X < 900) = P(-0.4 < zX < 1.6)

= P(zX < 1.6) - P(zX < -0.4)
= 1 - P(zX < -1.6) - P(zX < -0.4)
= 1 - 0.0548 - 0.3446
= 0.6006