## 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.

## 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 PopulationZ-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.

## 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.