How to Find a Z-Score with the Z-Score Formula

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.

I'll show you How to Find a Z-Score by using the z-score formula. I'll guide you through a couple examples and give and important calculator tip.

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), $ {\mu}$ is the mean of the population, and $ {\sigma} $ is the standard deviation of the population.	$$z = \frac{x-\bar{x}}{s}$$ Where $x$ is the raw score (the data value), ${\bar{x}}$ is the mean of the sample, and $s$ is the standard deviation of the sample.

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),
$ {\mu}$ is the mean of the population,
and $ {\sigma} $ is the standard deviation of the population.

$$z = \frac{x-\bar{x}}{s}$$

Where $x$ is the raw score (the data value),
${\bar{x}}$ is the mean of the sample,
and $s$ is the standard deviation of the sample.

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.