# Relationship between Binomial and Normal Distributions

Theorem 1: If x is a random variable with distribution B(n, p), then for sufficiently large n, the following random variable has a standard normal distribution:

where

ProofClick here for a proof of Theorem 1, which requires knowledge of calculus.

Corollary 1: Provided n is large enough, N(μ,σ) is a good approximation for B(n, p) where μ = np and σ2 = np (1 – p).

Observation: The normal distribution is generally considered to be a pretty good approximation for the binomial distribution when np ≥ 5 and n(1 – p) ≥ 5.  For values of p close to .5, the number 5 on the right side of these inequalities may be reduced somewhat, while for more extreme values of p (especially for p < .1 or p > .9) the value 5 may need to be increased.

Example 1: What is the normal distribution approximation for the binomial distribution where n = 20 and p = .25 (i.e. the binomial distribution displayed in Figure 1 of Binomial Distribution)?

As in Corollary 1, define the following parameters:

Since np = 5 ≥ 5 and n(1 – p) = 15 ≥ 5, based on Corollary 1 we can conclude that B(20,.25) ~ N(5,1.94).

We now show the graph of both pdf’s to see visibly how close these distributions are:

Figure 1 – Binomial vs. normal distribution

### 9 Responses to Relationship between Binomial and Normal Distributions

1. Kindly give the full information about Binomial and Normal Distribution.

• Charles says:

You can read much more about the binomial and normal distributions on the website. For example, please see the webpages
http://www.real-statistics.com/normal-distribution/
http://www.real-statistics.com/binomial-and-related-distributions/binomial-distribution/
Charles

• elasfar says:

3. In a particular road it is estimated that there is a 25% chance that any specified house will be burgled over a period of two years, independently for each house. There are seven houses in the road. Find the probability that fewer than two houses will be burgled over the period. how i can solve this please.

• Charles says:

Elasfar,
You have a situation that matches the requirements of the binomial distribution. What you are looking for is the probability that either 0 houses will be burgled or 1 house will be burgled. See the following webpage for more help:
Binomial Distribution
Charles

2. Gagan says:

Thank you for the information!

3. Sonya says:

Thank you for the clear explanations!
I was wondering if there is a standard (peer reviewed?) reference for the observation that the normal distribution is a good approximation for the binomial distribution when n > 10 and .4 < p 30 and .1 < p < .9. That would be very helpful!

• Charles says:

Sonya,
I was not able to find the reference to this, but I have now checked the statement against real data. When n > 10 and .4 < p < .6, the approximation is pretty good. However, for p near .1 or .9, the approximations weren't that good for n near 30. I have now changed the wording on the referenced webpage. Charles

4. Florence says:

Thank you so much. This is a good tutorial.

5. Riddhima says:

Great explanation!