Z-score and percentile from value, mean and standard deviation — with interpretation.
The z score formula counts how many standard deviations a value sits from the mean. The percentile that follows assumes a normal (bell-shaped) distribution.
Z score
How to use this tool
Type the value you want to place, the mean of its distribution and the standard deviation — for example, a score of 85 with mean 70 and SD 10.
The headline z score is signed: positive means above the mean, negative means below it. Its size in standard deviations is stated alongside, in plain words.
The percentile tells you what share of a normal distribution falls below your value, and the 68–95–99.7 band tells you how unusual it is. Both assume a bell-shaped distribution.
Copy the z score, percentile and working for a statistics answer or a report.
Formula & worked example
Worked example: a test score of 85, in a distribution with mean 70 and standard deviation 10, gives a z score of (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5. That means the score sits 1.5 standard deviations above the mean. Under the normal curve, about 93.32% of values fall below a z score of 1.5, so 85 is at the 93.32nd percentile and only about 6.68% of scores are higher. A score equal to the mean (70) has z = 0 and sits at the 50th percentile.
Limitations — read this before you rely on it
The percentile assumes a normal (bell-shaped) distribution; for skewed or bimodal data it is only a rough guide, and the z score itself should be quoted instead. The standard deviation must be greater than 0 — if every value is identical there is no spread to measure against. This calculator uses the population-style z formula with the standard deviation you supply; if you computed a sample standard deviation, make sure it is the one your question expects, since the two differ slightly.
Frequently asked questions
A z score counts how many standard deviations a value is from the mean: z = (value − mean) ÷ standard deviation. A z score of 1.5 means 1.5 standard deviations above the mean; −1.5 means 1.5 below it. It lets you compare scores from different tests, scales or groups on one common scale.
Use the z score formula: z = (x − μ) ÷ σ, where x is your value, μ is the mean and σ is the standard deviation. For 85 with mean 70 and SD 10: (85 − 70) ÷ 10 = 1.5. This calculator does it and converts the result to a percentile for you.
The percentile is the share of a normal distribution below your value, found from the normal CDF (the area under the bell curve up to your z score). Z = 0 gives the 50th percentile, z = 1 gives about the 84.13th, z = 1.5 gives about the 93.32nd and z = −1 gives about the 15.87th.
Roughly 68% of values fall within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3 — the empirical rule. So a z score beyond ±2 is unusual and beyond ±3 is very unusual, in a normal distribution.
Only if the data are (approximately) normally distributed — bell-shaped and symmetric. For skewed data, the percentile from the normal curve is misleading: a value can have a high z score without truly being at that percentile. The z score itself is still a valid standardised distance; it is the percentile that needs the normality assumption.