Z-Score Calculator
Standard deviations from the mean, and the percentile that follows.
Work out Z-Score. Standard deviations from the mean, and the percentile that follows. States the assumption it rests on, rather than hiding it.
Converting a z-score to a percentile assumes a normal distribution. Income, house prices and response times are all strongly skewed, and for those the z-score is still correct while the percentile it implies can be badly wrong.
z-score
+2.000
2.00 standard deviations above the mean
A z-score of +2.000 says the value sits 2.00 standard deviations above the mean. The percentile that follows — 97.7% — is the part that carries an assumption: it is only meaningful if the data is roughly normally distributed. The z-score itself is always valid; the percentile is not.
How the Z-Score Calculator works
A z-score says how far a value sits from the mean, measured in standard deviations: z = (x − μ) / σ. An IQ of 130 against a mean of 100 and a standard deviation of 15 is exactly two — which is why 130 is the number people quote. The percentile that follows is the part worth being careful about, because it assumes the data is normally distributed, and plenty of real data is not.
Also known as: standard score calculator · z score to percentile · how to calculate z score · z score formula
The calculation itself
z = (x − μ) / σ. Subtract the mean, divide by the standard deviation, and the result is a distance measured in standard deviations rather than in the original units.
That rescaling is the whole point. It makes a test score and a blood pressure reading comparable, because both become 'how far from typical, in units of typical spread'.
In practice
An IQ of 130 against a mean of 100 and a standard deviation of 15 gives (130 − 100) ÷ 15 = exactly 2.00. That is why 130 is the number people quote — it is the round z-score, not a round IQ.
The percentile follows from the normal distribution: a z of 2 sits at the 97.72nd percentile, so 2.28% of the population scores higher. The two-tailed p-value is 0.0455, which is the figure a significance test would report.
A value below the mean gives a negative z, and the sign is doing real work: −1.5 and +1.5 are equally unusual, in opposite directions.
Where this goes wrong
The z-score is a rescaling and is always arithmetically valid. The percentile is not — it assumes the data follows a normal distribution, and a great deal of real data does not.
Income is the standard example. It is strongly right-skewed, so an income two standard deviations above the mean is nowhere near the 97.7th percentile; the long tail of very high earners inflates the standard deviation and the normal curve gets the shape wrong. House prices and response times behave the same way.
The rule of thumb: trust the z-score, question the percentile, and ask what the distribution actually looks like before quoting one.
Frequently asked questions
How do I calculate a z-score?
Subtract the mean from your value, then divide by the standard deviation. An IQ of 130 with a mean of 100 and a standard deviation of 15 gives (130 − 100) ÷ 15 = 2.00.
What is a good z-score?
There is no good or bad in the abstract — it depends entirely on what is being measured. A z-score of +2 is high, reached by about 2.3% of a normal population. A z-score of −2 is equally unusual in the other direction.
How do I convert a z-score to a percentile?
Through the standard normal cumulative distribution, which this page computes. A z of 2 corresponds to the 97.7th percentile, meaning 97.7% of the distribution falls below that value. This step, not the z-score itself, is where the normality assumption enters.
Can a z-score be negative?
Yes, and it simply means the value is below the mean. The sign carries the direction and the magnitude carries the distance, so −1.5 and +1.5 are equally far from typical.
What does a z-score of 0 mean?
The value is exactly the mean, which puts it at the 50th percentile. Half the distribution falls below it.
Is a z-score valid for skewed data?
The z-score itself is, since it is only a rescaling. The percentile is not: for income, house prices or response times — all strongly skewed — the implied percentile can be badly wrong even though the z-score is arithmetically correct.
Put this calculator on your own site
Free to use, on any site, commercial or not. Paste this where you want it to appear. It is a plain iframe, so it works in WordPress, Squarespace, Wix, Webflow, Ghost and anything else that accepts HTML.
<iframe src="https://www.thecalclibrary.com/embed/z-score-calculator" width="100%" height="640" style="border:1px solid #e2e8f0;border-radius:12px" loading="lazy" title="Z-Score Calculator"></iframe>The only condition is that the credit line stays visible. It sits inside the frame, so you do not have to do anything to keep it.
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