Normal Approximation to Binomial

The Normal distribution can be used to approximate Binomial probabilities when n is large and p is close to 0.5. In answer to the question "How large is large?", or "How close is close?", a rule of thumb is that the approximation should only be used when both np>5 and nq>5. Click on the link below to load an Excel spreadsheet that allows you to check how well the Normal distribution fits for values of n up to 1000.

Load spreadsheet

Example

Suppose that in the biro example (5% defective) we wanted to know the probability of getting less than 40 defectives in a bulk purchase of 100 packets - i.e. 1000 biros. Again, we must assume that this purchase represents a random sample of all biros produced.

For the approximation to work we must consider three things:

we must match the mean of the Normal distribution to the mean of the Binomial
- otherwise the Normal curve will be centred in the wrong place,
we must match the standard deviation of the Normal to that of the Binomial
- otherwise the Normal curve will not be the correct width
we must make an ajustment to take account of the fact that the Binomial variable is discrete while the Normal variable is continuous - the continuity correction.

normal.gif (9988 bytes)

The mean number of defectives in 1000 biros = np = 1000 x 0.05 = 50
The standard deviation = sqrt(npq) = sqrt(1000 x 0.05 x 0.95) = 6.892.

Less than 40 defectives means 39 or less on the discrete scale, but 39 extends up as far as 39.5 on the continuous scale. So the approximation is the area under the Normal curve below 39.5.

Hence, Pr(defectives < 40) = Pr(defectives <39.5) ~ Pr( Z < [39.5 - 50]/6.892 ) = Pr( Z > -1.524)
which, from tables of the Normal distribution, is 0.0638.

Using the Binomial formula the corresponding probability would have been 0.0598.

You have now completed the recommended route through this unit on the Binomial Distribution. Make sure you visit our Links page to find out about other online resources relevant to this topic.