Z-Score Calculator

How to calculate a z-score

A z-score says how many standard deviations a value lies from the mean:

z = (x − μ) ÷ σ

where x is the value, μ the mean and σ the standard deviation. With x = 75, μ = 70 and σ = 5, z = 5 ÷ 5 = 1. A value of 62 on the same scale gives z = (62 − 70) ÷ 5 = −1.6: negative, because it is below the mean. Z-scores have no units, so they let you compare results measured on different scales — a 75 on one test and a 640 on another — by how unusual each is.

If you have the raw data rather than the mean and standard deviation, work those out first with the standard deviation calculator, then come back with the numbers.

From a z-score to a probability

For data that follow a normal distribution, the z-score fixes the share of values on either side. That share is the area under the standard normal curve, written Φ(z):

The area to the left is the percentile: z = 1 sits at about the 84th percentile, z = 2 at about the 98th. The last column is the empirical rule — about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. The calculator draws the curve and shades the area each result refers to, which is the quickest way to check that you have the tail you meant.

From a probability back to a z-score

Choose Z ↔ probability and say which area you know. A left-tail area of 0.95 gives z = 1.645; a central area of 0.95, or a two-tailed p-value of 0.05, gives z = ±1.96. These are the critical values used in confidence intervals and hypothesis tests: 1.645 for 90% confidence, 1.96 for 95% and 2.576 for 99%. The result also lists every other area for the same z, so one entry gives the whole row of a z-table.

Area between two z-scores

To find the share of values in a range, convert both ends to z-scores and use Between two z. Between z = −1 and z = 2 lies 0.8186 of a normal distribution, so about 82% of scores fall between one standard deviation below and two above the mean. The tails outside are reported too: 0.1587 on the left and 0.0228 on the right.

When the z-score is not enough

The probabilities assume the data are close to normal. Heights, test scores and measurement errors often are; incomes, waiting times and counts usually are not — they are skewed, and a z-score of 2 may be much more or less rare than 2.3%. With small samples, a t-distribution replaces the normal curve for tests on the mean. To see how far apart two numbers are in percentage terms instead, use the percentage calculator; for powers, roots and logarithms, the scientific calculator.