A z-score is how many standard deviations a value sits from the mean: z = (x − mean) / SD. It standardizes values from any normal distribution onto one comparable scale.

What does a negative z-score mean?

A negative z-score means the value is below the mean — the more negative, the further below. A z-score of −2 is two standard deviations below the mean, which under a normal distribution puts it around the 2nd percentile.

How do I convert a z-score to a percentile?

A z-score's percentile is the area under the standard normal curve to its left — the share of values that fall below it. Enter the z-score as the score with mean 0 and standard deviation 1, and this calculator computes that area for you.

Z-Score Calculator

Free z-score calculator: enter a score, mean, and standard deviation to get the z-score and percentile instantly, with the formula shown. Works as a z-table replacement too.

Direction Mean Standard deviation

Score

z = 1.5, percentile = 93.3193, score = 85

Z-score, step by step

z = (x − mean) / SD = (85 − 70) / 10 = 1.5.

The percentile is the area under the normal curve to the left of that z-score: 93.3193.

What a z-score is

A z-score measures how many standard deviations a value sits from the mean of its distribution: z = (x − mean) / SD. A z-score of 0 is exactly at the mean; positive z-scores are above the mean, negative ones are below. Because it standardizes any normal distribution onto the same scale, a z-score lets you compare values that come from different distributions — a test score, a height, a measurement — using one consistent unit.

Reading the sign and size

The sign tells you direction: a z-score of −1.5 is one and a half standard deviations below the mean; +2 is two standard deviations above. The magnitude tells you how unusual a value is — under a normal distribution, most values (about 68%) fall within one standard deviation of the mean (z between −1 and 1), and values beyond z = ±3 are rare. A z-score's percentile — the share of the distribution below it — turns that abstract distance into a concrete rank: a z-score of 1.5 sits at about the 93rd percentile, meaning roughly 93% of values in the distribution fall below it.

Worked example

Take a test with a mean score of 70 and a standard deviation of 10, and a student who scored 85. Their z-score is (85 − 70) / 10 = 1.5 — one and a half standard deviations above the mean. Looking up that z-score on the standard normal distribution gives a percentile of about 93.32%, meaning the student scored better than roughly 93% of the class. Leaving the mean at 0 and the standard deviation at 1 turns this into a plain z-table lookup: entering 1.5 as the score with those defaults gives the same z = 1.5 and percentile ≈ 93.32%.

The empirical rule

The 68–95–99.7 rule is a fast way to sanity-check a z-score's percentile: about 68% of values fall within one standard deviation of the mean (z between −1 and 1), about 95% within two, and about 99.7% within three. So a z-score of exactly 1 should land around the 84th percentile (half of the remaining 32% below the mean's outer 68%, plus the 50% below the mean itself), and a z-score of 2 should land around the 98th. If a percentile this tool reports looks far off from that rule of thumb, double-check the mean and standard deviation you entered.

How to use it

Choose the direction: score → z-score and percentile, or percentile → z-score and score.
Enter the mean and standard deviation of the distribution — leave them at 0 and 1 for a plain standard-normal (z-table) lookup.
Enter the score (or the percentile, in the reverse direction).
Read the z-score, percentile, and the step-by-step working below the shaded curve.
Download the chart as PNG, SVG, or PDF, or export the numbers as CSV.

FAQ

What is a z-score?: A z-score is how many standard deviations a value sits from the mean: z = (x − mean) / SD. It standardizes values from any normal distribution onto one comparable scale.
What does a negative z-score mean?: A negative z-score means the value is below the mean — the more negative, the further below. A z-score of −2 is two standard deviations below the mean, which under a normal distribution puts it around the 2nd percentile.
How do I convert a z-score to a percentile?: A z-score's percentile is the area under the standard normal curve to its left — the share of values that fall below it. Enter the z-score as the score with mean 0 and standard deviation 1, and this calculator computes that area for you.