Z-Score Calculator

What Is a Z-Score Calculator?

A Z-Score Calculator determines the z-score, also called a standard score, for a value based on its mean and standard deviation. The result shows how far the value is from the mean when the distance is measured in standard deviations.

Z-scores are commonly used in statistics, mathematics, data analysis and education to compare values within a distribution. A z-score can quickly show whether a value is above or below the mean and how unusual its distance from the mean is in standard-deviation terms.

Why Use a Z-Score Calculator?

This free online calculator performs the standard z-score calculation for you, reducing the need to calculate the formula manually and making it easy to check statistical calculations.

How to Use the Z-Score Calculator

Enter the three values required by the z-score formula and the calculator will produce the standard score automatically.

1. Enter the data value — Enter the value for which you want to calculate the z-score.
2. Enter the mean — Enter the mean of the relevant data set or distribution.
3. Enter the standard deviation and choose precision — Enter a standard deviation greater than zero and select the number of decimal places you want to display.
4. Calculate and review the result — Select Calculate Z-Score to see the result and its interpretation.

How to Calculate a Z-Score

The standard formula for calculating a z-score is z = (x − μ) / σ. In this formula, x represents the individual data value, μ represents the population mean, and σ represents the population standard deviation. First subtract the mean from the value, then divide the difference by the standard deviation.

For example, if a value is 85, the mean is 75 and the standard deviation is 5, the calculation is (85 − 75) / 5, producing a z-score of 2. This means the value is two standard deviations above the mean.

What Does a Z-Score Mean?

A z-score describes the position of a value relative to the mean in standard-deviation units. A result of 0 means the value is exactly at the mean. A positive result means the value is above the mean, while a negative result means it is below the mean. The absolute size of the z-score indicates the distance from the mean.

Positive and Negative Z-Scores

Positive and negative z-scores indicate the direction of a value relative to the mean. For example, a z-score of 1.5 means the value is 1.5 standard deviations above the mean, whereas a z-score of −1.5 means it is 1.5 standard deviations below the mean. The same formula is used when working with a sample mean and sample standard deviation; the sample statistics are simply used as the mean and standard deviation inputs.

Understanding Z-Score Results

A z-score of 0 means the value is equal to the mean. A positive z-score indicates that the value is above the mean, while a negative z-score indicates that it is below the mean. The magnitude of the z-score represents the number of standard deviations separating the value from the mean.

For example, a z-score of 1 means the value is one standard deviation above the mean, while a z-score of −2 means the value is two standard deviations below the mean. A z-score describes position in standard-deviation units, but it does not by itself provide a percentile unless assumptions about the underlying distribution are made.

Frequently Asked Questions