A z-score tells you how many standard deviations a value is above or below the mean. A z-score of 0 means the value equals the mean; a z-score of 1 means it is one standard deviation above the mean.

What is a good or bad z-score?

It depends on context. In a normal distribution, about 68% of values have z-scores between -1 and +1, 95% fall between -2 and +2, and 99.7% between -3 and +3. Values beyond ±3 are considered statistical outliers.

Can the standard deviation be zero?

No. If all values in a data set are identical, the standard deviation is zero and the z-score is undefined (division by zero). This calculator will show an error in that case.

How is the percentile calculated?

The percentile is derived from the cumulative distribution function (CDF) of the standard normal distribution. It represents the percentage of values in a normal distribution that fall at or below the given z-score.

What is the difference between population and sample standard deviation?

Population standard deviation (σ) uses N in the denominator; sample standard deviation (s) uses N-1. Use population SD when you have data for the entire group, and sample SD when working with a subset. The z-score formula is the same for both.

Z Score Calculator

Name: Z Score Calculator
Availability: InStock
Author: Nham Vu

Enter a value, mean, and standard deviation to instantly calculate the z-score and see where that value falls on the normal distribution.

Enter Values

Value (x)

The data point you want to evaluate.

Mean (μ)

The average of the population or sample.

Standard Deviation (σ)

Must be greater than zero.

Formula

z = (x − μ) / σ

x = observed value • μ = mean • σ = standard deviation

Fill in the fields on the left and click Calculate to see your z-score.

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Summary

Enter a value, mean, and standard deviation to instantly calculate the z-score and see where that value falls on the normal distribution.

How it works

Enter the observed value (x) you want to evaluate.
Enter the population mean (μ) for your data set.
Enter the standard deviation (σ), which must be greater than zero.
The calculator applies the formula z = (x - μ) / σ.
The result shows the z-score, its interpretation, and the approximate percentile.

Use cases

Determine how unusual a test score is compared to the class average.
Compare measurements from different data sets with different units.
Identify outliers in a data set (z-scores beyond ±2 or ±3).
Convert raw scores to standardized scores for statistical analysis.
Evaluate performance metrics against historical averages.
Assess financial returns relative to market benchmarks.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu