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Z-Score Calculator

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A z-score says how many standard deviations a value sits from the mean. What you usually want is the probability attached to it — the percentile, the tail area, whether it clears a threshold — and that is what this gives you, in whichever direction you need it.

What do you have?

How to use the z-score calculator

  1. 1Choose what you are starting from: a raw value, a z-score, or a probability.
  2. 2Fill in the fields that appear. The mean and standard deviation are optional in the last two modes.
  3. 3Read the percentile and the three tail probabilities.
  4. 4Compare your score against the critical value table if you are testing a hypothesis.

Examples

A test score against the class

Input
Value 82, mean 74, standard deviation 6
Result
z = 1.333 · 90.88th percentile · two-tailed p = 0.1824

The 5% critical value

Input
Probability 0.05, two-tailed
Result
z = ±1.9600 — the figure behind the familiar 95% confidence interval

About the z-score calculator

What standardising is for

Subtracting the mean and dividing by the standard deviation strips a measurement of its units. A height in centimetres, a test score out of 200 and a reaction time in milliseconds all become plain numbers on the same scale, and can then be compared directly.

That is the whole trick behind a z-score: not that it tells you something new about the value, but that it puts values from different distributions into a single currency. It is why the same table of critical values works for every normally distributed quantity there is.

The empirical rule, and its limits

About 68% of a normal population falls within one standard deviation of the mean, 95% within two and 99.7% within three. The exact figures are 68.27%, 95.45% and 99.73%, and the familiar 1.96 rather than 2 is what gives exactly 95%.

The rule holds only for a normal distribution. Income, city populations and file sizes are all strongly skewed, and applying the rule to them produces confident nonsense — a distribution with a long right tail will have far more than 0.15% of its mass three deviations above the mean.

Frequently asked questions

Should I use z or t?
z, when the population mean and standard deviation are known or the sample is large. t, when you have only a sample standard deviation and n is small. The difference is real at small samples: the two-tailed 5% critical value is 1.96 for z but 2.26 at n = 10 under t. This page computes the normal case and does not disguise a t problem as a z one.
What does the two-tailed p actually mean?
It is the probability of seeing a value at least this far from the mean in either direction, if the null hypothesis were true. It is not the probability that the hypothesis is false, and it is not the probability the result happened by chance — both of those are common misreadings that reverse what is being conditioned on.
Why does the tool say a significant result is not necessarily an important one?
Because clearing a threshold and mattering are different questions. With a large enough sample almost any difference becomes statistically significant, including differences too small to act on. The size of the effect, and whether it is worth anything, is a separate judgement the arithmetic cannot make for you.
Where do the probabilities come from?
From the normal cumulative distribution function, computed through a complementary error function with a refinement step, accurate to well under a part in a billion across the range this page uses. It is the same implementation the sample size calculator on this site relies on, and it is checked against published values in the test suite.

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