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Z-Score Calculator (With Percentile)

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Z-Score Calculator

A z-score tells you how many standard deviations a value sits above or below the mean: z = (x − μ) ÷ σ. Enter a raw score with the mean and standard deviation of its distribution and this calculator returns the z-score, the area to the left (the percentile) and the area to the right, using the standard normal distribution.

Z-scores make different scales comparable. A student who scored 85 on a test with mean 70 and SD 10 has z = 1.5, which is a stronger result than a 90 on a test with mean 80 and SD 15 (z = 0.67). For test scores such as the SAT, ACT or IQ scales, the percentile that comes with the z-score is usually what admissions or reports quote.

The steps show the substitution into the formula and the table look-up, so you can reproduce the answer with a printed z-table. To go the other way, from a percentile to a score, use the inverse normal calculator.

Result and step-by-step solution

z-score
1.5

The score 85 is 1.5 standard deviations above the mean.

Percentile (area to the left)93.32%
P(Z < 1.5)0.9332
P(Z > 1.5)0.0668
P(|Z| > 1.5)0.1336
  1. Formula
    z = (x − μ) ÷ σ
  2. Substitute
    z = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5
  3. Look up the standard normal table
    Φ(1.5) = 0.9332, so about 93.3% of values fall below 85 and 6.7% fall above it.

Frequently asked questions

What does a negative z-score mean?
The value is below the mean. A z-score of −1 is one standard deviation below the mean, and about 16% of a normal distribution lies below it.
How do I turn a z-score into a percentile?
Look up the area to the left of z in the standard normal table and multiply by 100. For z = 1.5 the area is 0.9332, so the score is at the 93rd percentile.
Is a z-score of 2 unusual?
Only about 2.3% of a normal distribution lies above z = 2, so yes, it is in the top few percent. Values beyond ±3 are rare (about 0.3% in total).
Do I use the sample or population standard deviation?
Use whichever describes the distribution the score belongs to. For standardised tests the published population mean and SD are used; for classroom data use the sample SD of the class.

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Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.

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