Normal Approximation to the Binomial Distribution (Year 13)

Under certain conditions, a binomial distribution (which is discrete) can be well-approximated by a normal distribution (which is continuous).

This approximation works well when:

• n is large, and

• p is not too close to 0 or 1 (ideally p ≈ 0.5).

More precisely, we often use the rule of thumb:

np ≥ 5 and n(1−p) ≥ 5.

Using the Approximation

For n large and p close to 0.5, we can approximate as follows

So the mean and variance, μ = np and σ² = np(1-p), for the binomial distribution are used as the parameters of the normal.

Why We Need a Continuity Correction

The binomial distribution is discrete: X=0, 1, 2, …, n.

The normal distribution is continuous: values run over an interval.

So when switching from binomial probabilities to normal probabilities, we adjust for this by using a continuity correction.

In general,

Visual Interpretation

Consider P(X ≦ 2) on the discrete scale i.e. the area of the blocks for X = 0, 1, 2. However 2 extends up to 2.5 on the continuous scale, so P(X ≦ 2) ≈ P(X ≦ 2.5)

Similarly, P(X = 3) ≈ P(2.5 ≦ X ≦ 3.5)

Example: Using Continuity Correction

A Final Reminder

For the normal distribution there is no distinction between < and ≤, > and ≥ because of continuity.