# Z-Score Calculator (with Formulas & Steps)

Enter the raw score, population mean, and standard deviation to find the z-score using the calculator below.

## Z-Score:

### Steps to Find the Z-Score

#### Use the Z-Score Formula

#### Substitute Values and Solve

z = ?

## On this page:

## How to Find the Z-Score

In statistics, the *z-score*, or *standard score*, is the number of standard deviations that a value is from the mean value. Each data point in a normal distribution has a z-score, which is the number of standard deviations away from the mean of the population.

Z-scores are a useful statistic because they allow you to calculate the probability of a score in a standard normal distribution, also known as the p-value. You can also use it to calculate the margin of error in survey data, for example.

### Z-Score Formula

Of course, the easiest way to find the z-score is to use the calculator above, but you can also find it using a formula. You can find the standard for any value in a normal distribution using the z-score formula.

z = x – μ / σ

Thus, the z-score *z* for a value *x* is equal to *x* minus the population mean *μ*, divided by the standard deviation *σ*.

### Steps to Find the Z-Score

You can find the z-score for a value in a few simple steps.

#### Step One: Find the Mean

The first step to finding the z-score is to find the mean for the population. You can use a mean calculator or use the mean formula.

μ = ∑x / n

The population mean *μ* is equal to the sum of all of the values in the population *x*, divided by the count of values in the population *n*.

#### Step Two: Find the Standard Deviation

The second step is to find the standard deviation for the data. You can use a standard deviation calculator or use the standard deviation formula.

The standard deviation *σ* is equal to the square root of the sum of squares divided by the population size *n*, where the sum of squares is equal to the sum of each value minus the population mean squared.

#### Step Three: Find the Z-Score

The final step to find the z-score is to use the formula above and substitute the values for the mean and standard deviation.

z = x – μ / σ

**For example,** let’s say Jane was taking a statistics course and she received a score of 93 on her final exam. The average score in the class was 89, and had a standard deviation of 3. Let’s calculate the z-score for Jane’s score.

z = Jane’s score – average score / standard deviation

z = 93 – 89 / 3

z = 4 / 3

z = 1.33

Thus, the z-score for Jane’s exam score is equal to **1.33**.

## How to Interpret a Z-Score

As noted above, the z-score is equal to the distance of a value from the mean in standard deviations, but what does that actually tell us? There are a few things we can take away from the z-score after we calculate it.

First, a positive z means that the raw score is greater than the mean, while a negative z means that the raw score falls below the mean. A z value of 0 means that the raw score is equal to the mean.

A very large z-score also tells us that the raw score is unusual, while a smaller z-score indicates that it might fall closer to the middle of the distribution. A z of greater than 3 or less than -3 generally indicates that the raw score is an outlier.

## What is a Z Table?

You might have also heard of a z table. A z table is a table that allows you to find the probability of a value being to the left of a z-score in a normal distribution.

A z table is also sometimes called a standard normal table.