## The Continuity Correction

I have illustrated the continuity correction for the binomial
distribution here, but exactly the same considerations apply when
approximating any discrete distribution by a continuous distribution.

The binomial distribution is a discrete distribution. That is, a
binomial random variable takes integer values. All the probability in
the binomial distribution sits in discrete lumps at the integers 0,
1, ..., n.

Look at the Bin(10, 0.5) distribution. The mean is 5 and the
variance is 2.5. How well does a N(5, 2.5) distribution approximate
the Bin(10, 0.5)?

The normal is a continuous distribution so we have to approximate
the lumps of probability at the integers by areas under the normal
curve. Take P(X = 3), for example. The exact binomial probability is
0.1172.

If we integrate the N(5, 2.5) density from 2 to 3 it is too low
everywhere and the result is too small.

F((3-5)/sqrt(2.5)) - F((2-5)/sqrt(2.5)) = F(-1.265) - F(-1.897)
= 0.1030 - 0.0289 = 0.0741

If we integrate the N(5, 2.5) density from 3 to 4 it is too high
everywhere and the result is too large.

F((4-5)/sqrt(2.5)) - F((3-5)/sqrt(2.5)) = F(-0.6325) - F(-1.265) = 0.2635 - 0.1030 = 0.1605

But if we integrate from 2.5 to 3.5 the result is an much closer
approximation.

F((3.5-5)/sqrt(2.5)) - F((2.5-5)/sqrt(2.5)) = F(-0.9487) - F(-1.581) = 0.1714 - 0.0569 = 0.1145

More generally, if we want to approximate the binomial probability
P(X <= a), we integrate under the normal density from minus
infinity to a+0.5. In the example below, a = 3 and the exact binomial
calculation gives 0.1719. The normal approximation (with continuity
correction) is

F((3.5-5)/sqrt(2.5)) = F(-0.9487) = 0.1714

If we want to approximate the binomial probability that P(X >=
a), we integrate the normal density from a-0.5 to infinity. In the
example below, a = 3 and the exact binomial calculation gives 0.9453.
The normal approximation (with continuity correction) is

1 - F((2.5-5)/sqrt(2.5)) = 1 - F(-1.581) = 1 - 0.0569 = 0.9431.

Last modified 1998-03-04