Z-Score Calculator

z = (x − μ) ÷ σ

x

The value being located — the measurement, the score, the reading you want to place on the scale

μ

The mean of the group the value belongs to. The subtraction is what turns an absolute value into a distance from the middle

σ

The standard deviation of that same group. Dividing by it is what makes the answer comparable across scales: a z-score of 1.5 means the same thing whether the data was in marks or millimetres. It must be greater than zero

x − μ

The distance from the mean, printed on its own as an absolute value so the gap can be read in the original units before it is rescaled

z

The z-score: negative when the value is below the mean, positive when it is above. The sign carries the direction, the size carries how unusual the value is

Use it to compare values that come from different scales, which is the whole reason the measure exists: a score of 85 in a class whose average is 75 is not the same achievement as 85 in a class averaging 90, and the z-scores say so directly. It is also the standard way of locating a single reading against a reference population, and the first step of most outlier rules. If you have the raw data rather than a mean and a standard deviation, get them from the standard deviation page first and come back — this page will not estimate them for you. Bear in mind what the band labels mean: the percentages behind them come from the empirical rule, the 68-95-99.7 rule, and that rule describes a normal distribution. Any value can have a z-score; only a normal one has 68% of its values inside one standard deviation.

1. A score one standard deviation above the class average

   1. Distance from the mean: 85 − 75 = 10 marks
   2. Divide by the standard deviation: 10 ÷ 10 = 1
   3. The z-score is 1, positive because 85 is above 75

   A z-score of exactly 1 lands in the first band, not the second: the empirical rule is a statement about the share of values falling within one standard deviation, so the boundary counts as inside, and 68% is the number attached to it. The ten marks of distance and the z-score of 1 are the same fact, one in the units of the exam and one in units of spread — that is exactly what the division buys you.

2. Exactly three standard deviations out

   1. Distance: 105 − 75 = 30
   2. Divide: 30 ÷ 10 = 3
   3. The band is the one that ends at 3, because the rule is about values beyond three standard deviations

   This is the boundary that matters most, and it is the one most often got wrong. Being exactly three standard deviations from the mean is not the same as being beyond three, and the 0.3% figure belongs to the beyond case. In practice it is a pedantic distinction — real measurements rarely land exactly on it — but it is the difference between the bands, and the table below spells out where each boundary sits.

3. Below the mean, where the sign flips

   1. Distance: 65 − 75 = −10, so the value is ten marks below the mean
   2. Divide: −10 ÷ 10 = −1
   3. The z-score is negative; its size is the same as in the first example, because the value is the same distance from the mean

   The sign is the direction and the size is the size. A value ten marks below the mean and a value ten marks above it have z-scores of −1 and +1, which are equally unusual — that is why the band is chosen from the absolute value of z, and why the panel reports the distance as a positive number. Read the two together: the sign says which side, the distance says how far.

A z-score is only as good as the mean and standard deviation you feed it: this page takes both as given and has no way to check them, so a mean typed from the wrong column makes every answer wrong without any sign of trouble. The percentages behind the bands describe a normal distribution, and real data often is not one — if the data is skewed, a large z-score may simply mean the distribution has a long tail rather than that the value is an outlier. There is also a small-sample trap worth knowing: NIST's Detection of Outliers points out that the largest possible z-score in a sample of n values is at most (n − 1) ÷ √n, so with fewer than about eleven values a z-score cannot reach 3 at all and a rule of thumb set at three standard deviations will never fire. Finally, the standard deviation must be greater than zero; a group in which every value is identical has no spread to measure against, and the page reports a failure rather than an answer.

What is the z-score formula?

z = (x − μ) ÷ σ: subtract the mean from the value, then divide by the standard deviation. A score of 85 against a mean of 75 with a standard deviation of 10 gives (85 − 75) ÷ 10 = 1. The subtraction puts the value on a scale centred on zero, and the division rescales it into units of spread, which is what makes two different scales comparable.

How many standard deviations is my value from the mean?

That is exactly what the z-score reports: its size is the number of standard deviations, and its sign says which side of the mean the value is on. A z-score of 2.5 means two and a half standard deviations above the mean; −0.4 means four tenths of a standard deviation below. The panel prints the raw distance as well, so you can read the gap in the original units next to the rescaled one.

What counts as a high z-score?

Under a normal distribution, about 68% of values fall within one standard deviation, 95% within two and 99.7% within three — the empirical rule. That makes a z-score beyond 2 uncommon and one beyond 3 rare, which is why three standard deviations is the usual line for a candidate outlier. Treat the numbers as a reading, not a verdict: they describe a normal distribution, and a skewed one can produce large z-scores for perfectly ordinary values.

Is a z-score the same as a percentile?

No, though they are related. A z-score is a distance in units of standard deviation; a percentile is a position in the ranked list of values. Converting one to the other needs the shape of the distribution — under a normal distribution a z-score of 1.645 sits at the 95th percentile, but that mapping only holds for a normal one. The z-score needs no assumption of its own, which is why it is reported here and the percentile is not.

What if the standard deviation is zero?

Then there is nothing to divide by, and the page reports a failure instead of a number. A standard deviation of zero means every value in the group is identical, so no value is any distance from the mean and the question has no answer. Infinities and errors printed as a result would be worse than saying so: a divide-by-zero here produces no meaningful reading, and the page would rather tell you that.

Can I use this for any data set, normal or not?

The z-score itself, yes: subtract the mean and divide by the standard deviation and you have a valid measure of relative position whatever the shape of the data. The band labels are the part that needs care. The 68-95-99.7 percentages are properties of a normal distribution, and a data set with a long tail will hand out z-scores well beyond 3 for values that are not really unusual for that distribution. Use the bands as a prompt to look at the value, not as a verdict on it.

• The Standard Normal Distribution — Introductory Statistics 2e, section 6.1 (the z-score as the number of standard deviations from the mean, and the Empirical Rule: about 68%, 95% and 99.7% within one, two and three standard deviations) — OpenStax, Rice University
• Detection of Outliers — e-Handbook of Statistical Methods, section 1.3.5.17 — National Institute of Standards and Technology (NIST)