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A z-score, or standard score, says how far a value sits from the mean, measured in standard deviations. It lets you compare values from different datasets on one scale. Enter a value, the mean and the standard deviation, and this tool returns the z-score and the percentile it corresponds to on the standard normal curve.
How is it calculated?
The formula
z = (x − μ) ÷ σ
where x is your value, μ (mu) is the mean and σ (sigma) is the standard deviation.
| z-score | Meaning |
|---|---|
| 0 | Exactly at the mean |
| +1 | One SD above the mean |
| −2 | Two SDs below the mean |
From z-score to percentile
The percentile is the area under the standard normal curve to the left of z — the proportion of a normal population that falls below your value. A z of 0 is the 50th percentile; +1 is about the 84th; +2 is about the 98th. This is the empirical “68–95–99.7” rule in action.
Why standardise?
Because it makes different scales comparable. A test score of 130 and a height of 190 cm mean nothing side by side — but their z-scores do, showing which is more unusual relative to its own distribution.
Worked example
Suppose IQ scores have a mean of 100 and a standard deviation of 15. A score of 115 gives z = (115 − 100) ÷ 15 = 1.0 — one standard deviation above the mean, at roughly the 84th percentile, meaning about 84% of people score below it. A score of 85 gives z = −1.0, the 16th percentile.
FAQ
What is a z-score?+
A z-score (standard score) is the number of standard deviations a value lies from the mean: z = (x − μ) ÷ σ. A positive score is above the mean, negative is below, and the size shows how far.
How do I calculate a z-score?+
Subtract the mean from your value, then divide by the standard deviation. For x = 70, mean = 60, SD = 5: z = (70 − 60) ÷ 5 = 2, meaning the value is two standard deviations above the mean.
What does the percentile mean?+
It’s the percentage of a normal distribution that falls below your value. A z-score of 0 is the 50th percentile (the mean); +1 is about the 84th; −1 about the 16th. This tool computes it from the normal curve.
What is a good or unusual z-score?+
By the 68–95–99.7 rule, about 68% of values fall within ±1 SD, 95% within ±2, and 99.7% within ±3. So a z beyond ±2 is fairly unusual, and beyond ±3 is rare.
Why use z-scores at all?+
They put values from different scales on a common footing. Standardising lets you compare, say, an exam mark to a height, or combine variables, because each is expressed in the same unit: standard deviations from its own mean.