**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:

Proof: Click 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 *σ ^{2} = 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**

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!

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

Thank you for the information!

Kindly give the full information about Binomial and Normal Distribution.

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

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.

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