In some cases, a Binomial Distribution can be approximated by a Normal Distribution (with the same mean and variance).

If X is a random variable with a Binomial Distribution we can write:

Then, if Y is a random variable with a Normal Distribution with the same mean and variance:

We can use two approximations. First, we can use the normal mass function: (in the applet this is the orange line)

A second approach is better and more practical if we use the tables of a normal distribution. Now we use a continuity correction, for example:

In the applet, this approximation is the red line.

In the applet, when the absolute error of this normal approximation is less than 0.005 we draw a blue rectangle. If the approximation is between 0.005 and 0.05 we draw a yellow rectangle and if the approximation is less accurate the rectangle is red.

Here we can see a case where the approximation seems accurate:

This approximation seems no so good:

In the next case the approximation is not accurate:

If n is large enough and p is "near" 0.5, then the skew of the distribution is not too great. In this case the normal distribution gives an excellent approximation. Because the normal approximation is not accurate for small values of n, there are several rules of thumb. For example, one is that we can use the normal approximation only if

The approximation generally improves as n increases and is better when p is not near 0 or 1.

Here we can see a typical example of the use of the normal tables to calculate probabilities in a Binomial distribution (using the correction for continuity adjustement)

With large n, exact calculations of a Binomial Distribution can be very time consuming. But using a table with the Normal Distribution is easy. Historically, it was the first use of the normal distribution, introduced by Abraham de Moivre.

LINKS

	Binomial distribution When modeling a situation where there are n independent trials with a constant probability p of success in each test we use a binomial distribution.
	Normal distribution The Normal distribution was studied by Gauss. This is a continuous random variable (the variable can take any real value). The density function is shaped like a bell.
	One, two and three standar deviations One important property of normal distributions is that if we consider intervals centered on the mean and a certain extent proportional to the standard deviation, the probability of these intervals is constant regardless of the mean and standard deviation of the normal distribution considered.
	Calculating probabilities in Normal distributions It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.
	Poisson distribution Poisson distribution is discrete (like the binomial) because the values that can take the random variable are natural numbers, although in the Poisson distribution all the possible cases are theoretically infinite.
	Student's t-distributions Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.
	Calculating probabilities in t Student distributions (Spanish)