5.5 - Approximating a binomial distribution

Normal approximation to a binomial distribution

For $X\sim\mathrm B(n,p)$, put $q=1-p$. Use a normal approximation when

Match the mean and variance with

The binomial mean and variance are in the formula booklet. The approximation conditions and the choice to use this normal model are not: learn them.

Worked example 1: a suitable approximation

Question. For $X\sim\mathrm B(100,0.4)$, justify a normal approximation and state its parameters.

Plan. Check both expected counts, then use the binomial mean and variance.

Answer. Both conditions hold, so use $Y\sim\mathrm N(40,24)$, with standard deviation $\sqrt{24}$.

Check. The proposed normal mean $40$ equals $\mathrm E(X)$, and its variance $24$ equals $\operatorname{Var}(X)$. The second parameter is not the standard deviation.

Worked example 2: one condition fails

Question. Is a normal approximation suitable for $W\sim\mathrm B(100,0.96)$?

Plan. Check both $np$ and $nq$, rather than just the larger expected count.

Answer. No: the expected number of failures is too small for the stated normal-approximation conditions.

Check. The mean is close to the maximum possible count $100$, so a symmetric bell curve would poorly reflect the short upper side and longer lower side. This agrees with the failed numerical condition.

Watch out

Do not write $\mathrm N(np,\sqrt{npq})$. Normal notation takes the variance as its second parameter. Use $\sqrt{npq}$ only when standardising.

Continuity correction

A continuity correction replaces an integer event boundary by the appropriate half-integer edge when using a continuous normal model for a discrete count.

Worked example 3: including the boundary count

Question. For $X\sim\mathrm B(100,0.4)$, write the continuity-corrected normal event approximating $\mathrm P(X\le42)$.

Plan. Include the whole unit bar centred at $42$, so use its upper edge.

The allowed counts end at $42$. Their bars extend up to $42.5$, giving

Answer. Use the normal area below $42.5$.

Check. The bar from $41.5$ to $42.5$ is included in full. Stopping at $42$ would include only half of the boundary bar.

Worked example 4: excluding the boundary count

Question. For the same binomial model, write the continuity-corrected normal event approximating $\mathrm P(X<42)$.

Plan. Translate $X<42$ to $X\le41$, then use the upper edge of the bar for $41$.

Answer. Use the normal area below $41.5$.

Check. The bar for $42$, from $41.5$ to $42.5$, is excluded. This matches the strict inequality in the original discrete event.