Z-Score Calculator

Z-score calculator at a glance#

A z-score, also called a standard score, tells you how many standard deviations a value sits above or below the mean. The formula is z = (x - mean) / SD: subtract the mean from your value, then divide by the standard deviation. A positive z-score is above the mean, a negative one is below, and zero means the value equals the mean.

Worked example: a test score of 85 in a group with a mean of 70 and a standard deviation of 10 has z = (85 - 70) / 10 = 15 / 10 = 1.5. So the score is 1.5 standard deviations above the mean.

This table uses a mean of 70 and a standard deviation of 10. Enter your own value, mean and standard deviation in the calculator above for the exact z-score, and to convert it to a probability. Use the population mean and standard deviation for population data, or the sample mean and standard deviation when you only have a sample.

Z-score of a sample mean#

When you have a sample rather than a single value, divide by the standard error instead of the standard deviation. The standard error is the standard deviation divided by the square root of the sample size, so the formula becomes z = (sample mean - population mean) / (SD / square root of n). For a sample mean of 115 against a population mean of 100 and a population standard deviation of 15 with n = 30, the standard error is 15 / square root of 30 = 2.739, so z = (115 - 100) / 2.739 = 5.48. A z-score that large means the sample mean is well outside the range you would expect by chance.

Reading a z-table#

A z-table, or standard normal table, gives the area under the normal curve to the left of a z-score, which is the probability of a value falling below it. The row gives the z-score to one decimal place and the column gives the second decimal. A z-score of 0 maps to 0.5, meaning 50% of values fall below the mean. A z-score of 1.96 maps to about 0.975, so 1.96 and -1.96 together enclose roughly 95% of a normal distribution.

Where z-scores are used#

A z-test checks whether a sample differs significantly from a known population when the population standard deviation is known, by comparing the z-score against a critical value. Prediction intervals use z-scores to set the range future values are likely to fall in at a given confidence level. In quality control, a z-score outside an accepted range flags a value as an outlier that may need attention.

Z-score FAQ#

What does a positive or negative z-score mean?#

A positive z-score means the value is above the mean, a negative one means it is below, and zero means it equals the mean. The number itself is how many standard deviations away the value sits.

How do you convert a z-score to a percentile?#

Look up the z-score in a z-table, which gives the area to its left as a decimal, then multiply by 100. A z-score of 1.0 maps to about 0.841, or the 84th percentile.

Can z-scores be used on data that is not normal?#

You can compute a z-score for any data, since it only needs a mean and a standard deviation. But the z-table probabilities assume a normal distribution, so read those with caution when the data is skewed.

What input formats does the calculator accept?#

The calculator takes data points one per line, separated by commas, separated by spaces, or a mix of these, so you can paste a column or a comma-separated list directly.