Normal Approximation to the Binomial
Prerequisites
Binomial
Distribution, History
of the Normal Distribution, Areas
of Normal Distributions
Assume you have a fair
coin and wish to know the probability that you would get 8 heads
out of 10 flips. The binomial distribution has a mean of μ
= Nπ = (10)(0.5) = 5 and a variance
of σ2 = Nπ(1π)=
(10)(0.5)(0.5) = 2.5. The standard deviation is therefore 1.5811.
A total of 8 heads is (8  5)/1.5811 =1.8973 standard deviations
above the mean of the distribution. The question then is, "What
is the probability of getting a value exactly 1.8973 standard
deviations above the mean?" You may be surprised to learn
that the answer is 0: The probability of any one specific point
is 0. The problem is that the binomial distribution is a discrete
probability distribution whereas the normal distribution is a
continuous distribution.
The solution is to round off and consider any value
from 7.5 to 8.5 to represent an outcome of 8 heads. Using this
approach, we figure out the area under a normal curve from 7.5
to 8.5. The area in green in Figure 1 is an approximation of the
probability of obtaining 8 heads.
The solution is therefore to compute this area.
First we compute the area below 8.5 and then subtract the area
below 7.5.
The differences between the areas is 0.044 which
is the approximation of the binomial probability. For these parameters,
the approximation is very accurate. The demonstration in the next
section allows you to explore its accuracy with different parameters.
You could find the solution using a table of the
standard normal distribution (a Z table) as follows:
 Find a Z score for 7.5 using the formula Z = (7.5  5)/1.5811
= 1.58.
 Find the area below a Z of 1.58 = 0.943.
 Find a Z score for 8.5 using the formula Z = (8.5  5)/1.5811
= 2.21.
 Find the area below a Z of 2.21 = 0.987.
 Subtract the value in step 2 from the value in step 4 to
get 0.044.
The same logic applies when calculating the probability
of a range of outcomes. For example, to calculate the probability
of 8 to 10 flips, calculate the area from 7.5 to 10.5.
