Z-Score Calculator
Calculate a z-score and percentile from a value, mean, and standard deviation.
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About this calculator
What Is a Z-Score?
A z-score, also called a standard score, tells you how many standard deviations a value lies above or below the mean of its distribution. It is a pure number with no units, which makes it possible to compare values from completely different scales — a height in centimeters and a test score, for instance. A positive z-score means the value is above the mean; a negative z-score means it is below; and a z-score of 0 means it equals the mean.
The Z-Score Formula
z = (x − μ) ÷ σ, where x is the raw score, μ is the population mean and σ is the population standard deviation. For a sample, the same formula uses the sample mean and sample standard deviation. If exam scores average 70 with a standard deviation of 8, a score of 86 has z = (86 − 70) ÷ 8 = 2 — two standard deviations above average.
Worked Example
A value of 85, with a mean of 75 and a standard deviation of 10, gives a z-score of 1 — exactly one standard deviation above the mean — which falls at roughly the 84th percentile.
Z-Scores and Probability
If a variable is roughly normally distributed, each z-score corresponds to a percentile and a probability, which is what a z-table shows. This calculator can convert between a z-score and a probability, and find the probability between two z-scores. Useful reference points:
- z = 0: the 50th percentile.
- z = 1: about the 84th percentile; about 68% of values fall between z = −1 and z = 1.
- z = 1.96: about the 97.5th percentile; 95% of values fall between −1.96 and 1.96.
- z = 3: about the 99.9th percentile; values beyond ±3 are rare.
What Z-Scores Are Used For
- Comparing results from different tests or measurements on a common scale.
- Spotting outliers: values beyond about ±2 or ±3 are unusual.
- Hypothesis tests and confidence intervals in statistics.
- Quality control and finance, where standard-score thresholds flag abnormal readings or returns.
Related Tools
You need the mean and spread first — find them with the Mean, Median, Mode Calculator and the Standard Deviation Calculator — and the Probability Calculator covers the normal distribution and combined events.
Worked example
A value of 85 with a mean of 75 and standard deviation of 10 gives a z-score of 1 — meaning it's exactly one standard deviation above the mean, at roughly the 84th percentile.
Frequently asked questions
What does a z-score tell you?
It measures how many standard deviations a value is from the mean of its distribution. A z-score of 0 is exactly average; positive values are above average, negative values are below.
How is percentile calculated from a z-score?
Using the cumulative distribution function of the standard normal distribution — it converts the z-score into the percentage of values expected to fall below it, assuming the data is normally distributed.
Does this work for data that isn't normally distributed?
The z-score itself is just a standardized distance from the mean and works regardless, but the percentile interpretation assumes a roughly normal (bell-curve) distribution — for heavily skewed data, the percentile figure may not be accurate.
