# How to Work with Normal Distributions

One of the most widely used and important continuous probability
distributions seen in statistics is the normal distribution. The normal
distribution is characterized by a bell shaped curve with the highest
point above the mean. The distribution is symmetrical and approaches the
horizontal axis but never touches it. But how do we work with normal
distributions? At the end of this article, you should be able to
calculate probabilities for a random variable that has a normal
distribution.

Suppose a random variable

Consider the following example:

Let

In symbols, we write this problem as P(8 ≤

Since the probabilities correspond with areas under the normal distribution curve, we can find the area under the curve from

For

Suppose we want to know the probability that a value chosen at random is greater than a certain value? Using the same value for mean and standard deviation, suppose we want the probability that

This is the same as one minus the probability of

This guide should give a student a basic understanding of how to calculate probabilities for a random variable that has a normal distribution.

Suppose a random variable

*x*has a normal distribution with a mean*m*and standard deviation*s*. To find areas and probabilities for such a random variable, convert*x*values to*z*using the formula*z*= (*x*-*m*)/*s*. Then use a table for the area of a standard normal distribution, found in most statistics textbooks, to find the corresponding areas and probabilities.Consider the following example:

Let

*x*have a normal distribution with*m*= 15 and*s =*4*.*Find the probability that an*x*value chosen at random from this distribution is between 8 and 19.In symbols, we write this problem as P(8 ≤

*x*≤ 19).Since the probabilities correspond with areas under the normal distribution curve, we can find the area under the curve from

*x*= 8 to*x*= 19. To do this, we convert the*x*values to*z*values.For

*x*= 8,*z*= (8 - 15)/4, therefore*z*= -1.75. For*x*= 19,*z*= (19 - 15)/4, therefore*z*= 1. Writing this in symbols, we get P(-1.75 ≤*z*≤ 1), This is the same as thinking of the area to the left of*z*= 1 minus the area to the left of*z*= -1.75. These values can be found using a table for the area of a standard normal distribution. Using the table, we get 0.8413 - 0.0401 = 0.8012.Suppose we want to know the probability that a value chosen at random is greater than a certain value? Using the same value for mean and standard deviation, suppose we want the probability that

*x*> 12?This is the same as one minus the probability of

*x*less than or equal to 12. In symbols, this is 1 - P(*x*≤ 12). Converting the*x*value to*z*value, we get*z*- (12 - 15)/4 = -0.75. The*z*value corresponding to -0.75 is 0.2266. Therefore, P(*z*> -0.75) = 1- .2266 = .7734.This guide should give a student a basic understanding of how to calculate probabilities for a random variable that has a normal distribution.