Z-Score Calculator (Standard Score and Percentile)
How many standard deviations a value is from the mean, with the percentile it corresponds to on a normal curve.
Inputs
- Working
- (85 − 75) ÷ 10 = 10 ÷ 10 = 1
- Meaning
- 1.00 standard deviation above the mean
- Percentile, if the data are normal
- 84.13% of values are below 85
- Share further from the mean (two-tailed)
- 31.73%
- Empirical rule
- within ±1σ, where about 68% of a normal distribution sits
Use this calculator on your own site
Free embed
- Two lines of HTML, unlimited views
- Shows a small CalculateEveryday bar, with a credit link under the calculator
What each Z-Score input means
- Value (x)
- The observation you want to place: a test score, a height, a measurement.
- Mean (μ)
- The mean of the distribution it comes from — the class average, the population mean.
- Std dev (σ)
- The standard deviation of that distribution. Use the population figure if you have it; whichever you use, use it consistently when comparing z-scores.
Z-Score formula
z = (x − μ) ÷ σ · percentile = Φ(z), the standard normal CDF · x = μ + zσ to go back
A z-score restates a value in units of standard deviations from the mean: subtract the mean, divide by the standard deviation. A z of 2 means two standard deviations above the mean, −0.5 means half a standard deviation below, 0 means exactly average. That makes scores from different scales comparable — a 620 on a test with mean 500 and SD 100 (z = 1.2) is a better result than 78 on a test with mean 70 and SD 10 (z = 0.8). If the data follow a normal distribution, the z-score also gives the percentile: the standard normal table says 84% of values lie below z = 1, 97.7% below z = 2. The empirical rule is the rough version — about 68% of normal data within one standard deviation, 95% within two, 99.7% within three — and the page says which of those bands the value falls in.
Z-Score worked example
A score of 85 where the mean is 75 and the standard deviation 10: (85 − 75) ÷ 10 = 1. The score is one standard deviation above the mean, which on a normal curve is the 84th percentile — 84% of scores are below it and 16% above. OpenStax's example works the same way: x = 17 with mean 5 and SD 6 gives (17 − 5) ÷ 6 = 2, two standard deviations up, the 97.7th percentile.
More Z-Score examples
A test score of 620 where the mean is 500 and the standard deviation 100: (620 − 500) ÷ 100 = 1.2, the 88.49th percentile if scores are normal.
A 62 on an exam with mean 70 and standard deviation 5: (62 − 70) ÷ 5 = −1.6, 1.6 standard deviations below the mean — only about 5.5% of a normal class scores lower.
Where the Z-Score calculator can give the wrong answer
- The percentile only holds for normal (bell-shaped) data. Incomes, house prices and waiting times are skewed, and a z of 2 in skewed data is not the 97.7th percentile — the z-score itself is still valid as a distance, the percentile is not.
- Which standard deviation you use matters. The population σ and a sample's s differ, and a z computed with one is not comparable with a z computed with the other; use the same convention throughout.
- The percentile comes from an approximation to the normal CDF accurate to about four decimal places — enough for any table lookup, not for extreme tails beyond z = 5.
- A z-score says where a value sits relative to its own distribution. It does not say the value is good or bad; a z of −2 on a golf score is excellent.
- For small samples the t-distribution, not z, is the right reference; that matters for confidence intervals more than for describing a single value.
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Z-Score reference table
| z | Percentile (share below) |
|---|---|
| −3 | 0.13% |
| −2 | 2.28% |
| −1.5 | 6.68% |
| −1 | 15.87% |
| −0.5 | 30.85% |
| 0 | 50% |
| 0.5 | 69.15% |
| 1 | 84.13% |
| 1.5 | 93.32% |
| 2 | 97.72% |
| 3 | 99.87% |
Z-Score FAQ
- Subtract the mean from the value and divide by the standard deviation: z = (x − μ) ÷ σ. A value of 130 with mean 100 and SD 15 has z = 30 ÷ 15 = 2. The page shows the subtraction and division for whatever you enter.
- The value is 1.5 standard deviations above (or below) the mean. On a normal curve that is about the 93rd percentile for +1.5 and the 7th for −1.5 — roughly 7% of values are further out on that side.
- Look it up in the standard normal table or use the normal CDF calculator on this site: z = 0 is the 50th percentile, z = 1 the 84th, z = 1.645 the 95th, z = 1.96 the 97.5th, z = 2.33 the 99th. This page shows the percentile for the z it finds.
- It depends on the direction that is good. For a test score, higher is better and z = 1 puts you in the top 16%. For blood pressure or a race time, lower is better. Roughly, |z| above 2 is unusual — outside 95% of normal data — and |z| above 3 is rare.