Z-Score Calculator

A z-score counts how many standard deviations a value is from the mean. Enter x, μ and σ to get z, its percentile, and the area to the left, to the right, in both tails or between two values, shaded on the bell curve.

z-score
1.5
Shaded area
0.9332
Percentile of x
93.32
z = 1.5

Step-by-step working

  1. Subtract the mean from the valuex − μ = 85 − 70 = 15
  2. Divide by the standard deviationz = 15 ÷ 10 = 1.5
  3. Look up the area to the left of z (standard normal table)P(Z < 1.5) = 0.9332 → the 93.3th percentile
  4. That left area is the proportion below x= 0.9332

Formula z = (x − μ) ÷ σ

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The bell curve in standard deviations

Every normal curve is the same shape once you measure in standard deviations. That’s what a z-score does: it counts how many SDs a value sits from the mean.

−1+1

68.27% of values lie within ±1 SD of the mean

That is the 68–95–99.7 rule (the empirical rule). Try a single z below: the shaded area to its left is the proportion of values below it, which is its percentile.

1.20

How to calculate a z-score

Subtract the mean from the value, then divide by the standard deviation. An exam with mean 70 and SD 10: a score of 85 has z = (85 − 70) ÷ 10 = 1.5. The area to the left of z = 1.5 is 0.9332, so 85 is about the 93rd percentile, assuming scores are roughly normal.

Positive z is above the mean, negative is below, 0 is exactly average. |z| > 2 happens about 5% of the time for normal data; |z| > 3 about 0.3%.

Comparing across different tests

z-scores put different scales on a common footing. An 85 on a test with mean 70, SD 10 (z = 1.5) is more impressive than a 90 on a test with mean 82, SD 8 (z = 1.0). To look up areas by hand, use the z-table.

Frequently asked questions

What is a good z-score?

It depends on context. For test scores, anything above 0 is above average; z = 1 is about the 84th percentile and z = 2 about the 98th.

Can I use z-scores if the data are not normal?

You can always compute z as a standardised distance, but turning z into a percentile or probability assumes a normal distribution.

How do I convert a z-score back to a value?

x = μ + z·σ. The normal distribution calculator’s inverse mode does this from a percentile.