Using the Normal Distribution to
Approximate the Binomial
Distribution
When np and n(1-p) are large
Why can We Use This?
We are claiming that you can use the continuous normal
distribution to make an estimation about the discrete binomial
distribution. How can this be?
Well, one nice thing we have on our side is the Central Limit
Theorem, which states that any distribution is approximately
normal if you take large enough samples.
Given this information, the normal model is a good
approximation to the binomial.
Conditions to be met
In order for this to happen, we need to make sure of two things:
1. np (or x) is greater than 5.
2. n(1-p) (OR N-X) IS GREATER THAN 5.
if these are met, we are good to use this normal approximation.
What is our new mean and standard
deviation?
The mean of the binomial distribution is np, and this is exactly
what we use in the normal approximation.
The standard deviation of the binomial distribution is the square
root of np(1-p), and yes, this is what we will use in our
approximation.
So, what’s next?
Continuity Correction
Most people like to make what is called a continuity correction,
which compensates for using a normal distribution to calculate a
discrete one.
This involves either adding or subtracting .5 from your value
before you standardize it. It depends on if you are asking for the
less than probability (add .5) or the greater than probability
(subtract .5).
Calculations
Just like the normal distribution, once we have a point, the mean,
and the standard deviation, we can calculate a Z-score.
Z = x-mean/(standard deviation)
X is the value asked in the problem, but make sure you do your
continuity correction first.
Example
Please refer to the video below for a good example on using the
normal to approximate the binomial.