Z-Score Calculator

A z-score is a distance measured in standard deviations

Take any value from a distribution, subtract the mean, and divide by the standard deviation. The result is how many standard deviations the value sits from the centre, and in which direction. A z of 2 means two standard deviations above the mean; a z of minus 0.5 means half a standard deviation below. That is the whole formula, and it is worth doing by hand once because every later idea in statistics, from confidence intervals to hypothesis tests, is a z-score in disguise.

The reason z-scores matter is that they put every normal distribution on one ruler. A test score of 85 on a test with mean 70 and standard deviation 6 is z = 2.5; a height of 190 cm in a population with mean 175 and standard deviation 6 is also z = 2.5. Both sit at the same percentile, about the 99.4th, and one table serves them both. This page reads that table for you, drawing the curve with the area below your value shaded, and reports the share above and the share within that distance of the mean as well.

The arithmetic here stays exact. A z of 15 over 6 is 5/2, and the page prints 5/2 with 2.5 beside it, rather than a decimal that would have to be rounded when the numbers are less convenient. The percentile, being an area under a curve, is necessarily a decimal, and it is labelled as one.

What to know about z-scores

• The sign tells you the side. Negative is below the mean, positive is above; a z of 0 is the mean itself.
• Almost everything sits within three. About 68% of a normal population is within one standard deviation, 95% within two, 99.7% within three. A z of 4 is genuinely rare.
• Percentile is the area to the left. A z of 1.28 sits at about the 90th percentile, 1.645 at the 95th, 2.33 at the 99th.
• The same formula runs backwards. The value at a given z is the mean plus z times the standard deviation, which is how cut-offs are set: the top 5% of a test starts at mean plus 1.645 sigma.
• A z-score assumes the distribution is roughly normal. The score itself is always computable; the percentile only means something when the bell shape holds.

How to find a z-score by hand

1. 1

   Subtract the mean from your value

   Keep the sign. A value below the mean gives a negative difference, and that sign carries through to the answer.

2. 2

   Divide by the standard deviation

   Use the population standard deviation if you have it, the sample one if that is all there is. Keep the result exact if the division is not clean.

3. 3

   Read the percentile from the normal curve

   The area to the left of z is the share of the population below your value. Tables list it; this page computes it directly.

4. 4

   For the reverse, multiply and add

   Value equals mean plus z times the standard deviation. This is how a percentile cut-off is turned back into a score.

Common z-scores and their percentiles

Area to the left of z on the standard normal curve.

Worked examples

x = 85, μ = 70, σ = 6

x = 48, μ = 50, σ = 2.5

z = 1.645, μ = 1000, σ = 150

Z-score mistakes

• Dividing by the variance instead of the standard deviation. The variance is the square; the answer would be off by a factor of sigma.
• Dropping the sign. A z of −1.5 and a z of 1.5 are at the 7th and 93rd percentiles, not the same place.
• Reading the wrong side of the table. Some tables give the area to the left, some the area between the mean and z, some the right tail. Know which one you are holding.
• Rounding z before looking it up. A z of 1.96 rounded to 2.0 moves the percentile from 97.5 to 97.7; on a cut-off that is real people.
• Using a z-score on a distribution that is nowhere near normal. The number still computes; the percentile is fiction.

Z-score FAQ