I’ll show you how to find a z-score by using the z-score formula. I’ll guide you through a couple examples that provide a mean, standard deviation, and raw score and ask you to find a z-score. For background information on z-scores, see What is a Z-Score? You’ll learn why we use z-scores. The Z-Score Calculator is a great resource. You’ll get a step-by-step solution so you can learn how to find a z-score on your own.

## Z-Score Formula

The z-score formula utilizes different symbols, depending on whether the data set under analysis represents a population or a sample. However, the general mathematics is the same for both instances.

Z-score Formula for a Population | Z-score Formula for a Sample |
---|---|

## $$ z = \frac{x-{\mu}}{\sigma} $$Where $ x $ is the raw score (the data value), | ## $$z = \frac{x-\bar{x}}{s}$$Where $x$ is the raw score (the data value), |

## Examples

Here are a couple examples that demonstrate how to find a z-score for a data value using the z-score formula.

**Example 1 – How to Find a Z-Score for an Income Raw Score**

The mean income for the population of residents in the city of Happy Town is **$75,000**. The standard deviation for this population of incomes is **$5000**. Mr. Miller, who lives in Happy Town, has an income of **$71,000**. What is the z-score for Mr. Miller’s income?

Reading the word problem, we see that **${\mu}$ is 75,000, ${\sigma}$ is 5000, and $x$ is 71,000**. First, substitute these values into the z-score formula for a population:

$$z = \frac{71,000-75,000}{5000}$$

To evaluate the formula, complete the subtraction in the numerator first, then divide that answer by the denominator.

$$z = \frac{-4000}{5000}$$

$$z = -0.8$$

**Example 2 – How to Find a Z-Score for a Minutes Raw Score**

The mean wait time for the sample of customers at a local post office is **4.5 minutes**. The standard deviation for this sample of wait times is **1.2 minutes**. Miss Brooklyn has been waiting at the post office for **7.8 minutes**. What is the z-score for Miss Brooklyn’s wait time?

Reading the word problem, we see that **$\bar{x}$ is 4.5, ${s}$ is 1.2, and $x$ is 7.8**. First, substitute these values into the z-score formula for a sample:

$$z = \frac{8-4.5}{1.2}$$

Next, remember to complete the subtraction in the numerator first, then divide that answer by the denominator.

$$z = \frac{3.5}{1.2}$$

$$z = 2.75$$

Z-scores do not have units. It would be incorrect to write that the answers to the two examples above were $-0.8$ or $2.75$ minutes. Rather, a z-score is a measure of the number of standard deviations the data value lies from the mean.

## Calculator Tip

If you are using a calculator to evaluate the z-score and you don’t put the numerator in parentheses, you’ll get the wrong answer!

$$z = {8-4.5/1.2}$$

**WRONG! **

By the mathematical order of operations, division would be evaluated before subtraction, giving you an incorrect answer. Make sure you enter the numerator in parentheses.

$$z = {(8-4.5)/1.2}$$

**CORRECT! **

## What’s Next?

Now that you understand how to find a z-score, check out the Z-Score Calculator. This calculator will both give an answer as well as a worked out solution for helping you learn how to solve the problem on your own. If you need more background meaning on z-scores, see What is a Z-Score.