# How to Find Probabilities for a Sample Mean

In statistics, you can easily find probabilities for a sample mean if it has a normal distribution. Even if it doesn’t have a normal distribution, or the distribution is not known, you can find probabilities if the sample size, *n*, is large enough.

The normal distribution is a very friendly distribution that has a table for finding probabilities and anything else you need. For example, you can find probabilities for

by converting the* *

to a *z*-value and finding probabilities using the *Z*-table (see below).

The general conversion formula from

Substituting the appropriate values of the mean and standard error of

the conversion formula becomes:

Don’t forget to divide by the square root of* n *in the denominator of *z*. Always divide by the square root of* n *when the question refers to the *average* of the* x-*values.

For example, suppose* X *is the time it takes a randomly chosen clerical worker in an office to type and send a standard letter of recommendation. Suppose* X *has a normal distribution, and assume the mean is 10.5 minutes and the standard deviation 3 minutes. You take a random sample of 50 clerical workers and measure their times. What is the chance that their average time is less than 9.5 minutes?

This question translates to finding

As* X *has a normal distribution to start with, you know

also has an exact (not approximate) normal distribution. Converting to *z,* you get:

So you want P(*Z* < –2.36).

Using the above *Z*-table, you find that P(*Z* < –2.36)=0.0091. So the probability that a random sample of 50 clerical workers average less than 9.5 minutes to complete this task is 0.91% (very small).

How do you find probabilities for

if* X *is* not *normal, or unknown? As a result of the Central Limit Theorem (CLT), the distribution of* X *can be non-normal or even unknown and as long as* n *is large enough, you can still find *approximate *probabilities for

using the standard normal (*Z*-)distribution and the process described above. That is, convert to a *z*-value and find approximate probabilities using the *Z*-table.

When you use the CLT to find a probability for

(that is, when the distribution of *X* is* not *normal or is unknown), be sure to say that your answer is an *approximation.* You also want to say the approximate answer should be close because you’ve got a large enough* n *to use the CLT. (If* n *is not large enough for the CLT, you can use the* t*-distribution in many cases.)