Z-Score Calculator

How to use the z-score calculator

1. Enter the value, the mean and the standard deviation. The mean and standard deviation should describe the group the value belongs to, such as all scores on the same test. If you only have raw data, get them from the standard deviation calculator first.
2. Read z. A positive z is above the mean, a negative z below it, and the note says how many standard deviations away it is.
3. Read the areas. The area below x is the value’s percentile if the data follows a normal distribution. The area above is the share of values higher than yours, and the two-sided figure is the share within the same distance of the mean in both directions.
4. Optionally enter a percentile to find the value that sits there, such as the score needed to be in the top 10% (the 90th percentile).

The z-score formula

The open textbook Introductory Statistics 2e from OpenStax (section 6.1) uses weight changes that follow a normal distribution with mean 5 and standard deviation 6, written N(5, 6). A value of x = 17 gives z = (17 − 5) ÷ 6 = 2: two standard deviations above the mean. A value of x = 1 gives (1 − 5) ÷ 6 = −0.67, about two-thirds of a standard deviation below. Another example in that section uses N(5, 2), where x = 10 gives z = 2.5.

The z-score itself is just arithmetic and works for any data. The percentiles are where the normal-distribution assumption comes in.

From z to a percentile

For a normal distribution, every z-score corresponds to a fixed share of the curve below it. These are the numbers printed in a standard normal table. The calculator computes them directly, and its values agree with the table in the NIST/SEMATECH e-Handbook of Statistical Methods to five decimal places:

• z = 0: 50% below (the mean is the median).
• z = 1: 84.13% below. The NIST table lists 0.34134 between 0 and 1, plus the 0.5 below 0.
• z = 1.96: 97.50% below, which is why 1.96 shows up in 95% confidence intervals.
• z = 2: 97.72% below, so only about 2.3% of values are higher.
• z = −1.5: 6.68% below.

The empirical rule from the same OpenStax chapter is the quick version: about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.

Comparing scores on different scales

The main use of a z-score is putting different measurements on a common scale. OpenStax makes this point with its two distributions: x = 17 under N(5, 6) and y = 4 under N(2, 1) both have z = 2, so each is equally far above its own mean.

A classroom example: you score 82 on a test where the class mean is 70 and the standard deviation is 8, and 88 on another where the mean is 80 and the standard deviation is 4. The raw 88 looks better, but the z-scores are (82 − 70) ÷ 8 = 1.5 and (88 − 80) ÷ 4 = 2.0. Relative to the class, the second result is stronger. Assuming normal scores, 1.5 is about the 93rd percentile and 2.0 about the 98th.

From a percentile back to a value

To find the cutoff for a percentile, look up the z for that percentile and convert back: x = μ + z × σ. The 90th percentile is at z = 1.2816. On the first test above (mean 70, standard deviation 8), that is 70 + 1.2816 × 8 = 80.25, so a score of about 80 or more is in the top 10%.

Some other useful z values: the 75th percentile is at 0.674, the 95th at 1.645, and the 99th at 2.326. Percentiles below 50 have negative z values; the 10th percentile is at −1.2816.

When the percentiles don’t apply

Real data is often not normal. Incomes, house prices and waiting times are skewed with long right tails, and test scores can bunch up near a maximum. For skewed data, the z-score is still a fair measure of distance from the mean, but the percentile from the normal curve can be well off. If you have the full data set, count the share of values below yours instead.

The mean and standard deviation should come from the same group as the value. Using a national average with your class’s standard deviation, for example, mixes two different distributions.

Values more than about three standard deviations from the mean are rare under a normal curve (about 0.3% combined). If you see several of them, check for data errors or a distribution with heavier tails.

Example: what share falls below a limit?

Suppose a filling machine puts a mean of 500 grams in each bag with a standard deviation of 4 grams, and bags under 492 grams are rejected. The limit is at z = (492 − 500) ÷ 4 = −2, and the area below z = −2 is 2.28%. If the fills are close to normal, roughly 23 bags in 1,000 would fail. Raising the mean to 504 grams moves the limit to z = −3, where only about 0.13% fall below.

One-sided and two-sided areas

The area above a positive z is a one-sided tail: the chance of a value at least that high. The chance of a value at least that far from the mean in either direction is twice as large. For z = 2, one tail holds 2.28% and both tails together 4.55%, which is why the two-sided “between” figure in the results is 95.45%. In hypothesis testing, these tail areas are the p-values for a z-test, so a z of 1.96 or more in either direction corresponds to a two-sided p-value of 0.05 or less.

Z-scores in a spreadsheet

In Excel, =STANDARDIZE(x, mean, sd) returns the z-score, =NORM.S.DIST(z, TRUE) the area below it, and =NORM.S.INV(0.9) the z for the 90th percentile (1.2816). Some fields rescale z-scores to avoid negative numbers. A T-score, for example, is 50 + 10z, so z = −0.67 becomes about 43.3 and z = 2 becomes 70. For a whole column of z-scores, compute the mean and standard deviation once in their own cells and refer to them with absolute references.

Sources & assumptions

How this calculator gets its numbers. Full method: our methodology.

• z = (x − μ) ÷ σ, with σ above 0.
• Percentiles assume a normal distribution; they come from the standard normal CDF computed with an error-function approximation accurate to about 1 part in 10 million.
• The value at a percentile uses the inverse normal CDF: x = μ + z × σ.

Sources

1. OpenStax (Rice University), Introductory Statistics 2e, 6.1 The Standard Normal Distribution. z = (x − μ) ÷ σ with N(5, 6): x = 17 gives z = 2 and x = 1 gives z = −0.67; N(5, 2): x = 10 gives z = 2.5; and the empirical rule (about 68%, 95% and 99.7% within 1, 2 and 3 standard deviations).
2. NIST/SEMATECH e-Handbook of Statistical Methods, Cumulative Distribution Function of the Standard Normal Distribution. Table of the area under the normal curve from 0 to z (0.34134 at z = 1, 0.47500 at z = 1.96, 0.47725 at z = 2), used to check the calculator’s percentiles.

Last reviewed November 2, 2026. For skewed data, count the share of values below yours instead of using the normal curve.