ZLab

Z-Score Calculator

z = (x − μ) / σ, plus the percentile it maps to.

Score x

Mean μ

Std dev σ

Z-score: 1.5
Percentile: 93.32%

z = (x − μ) / σ

What you entered

Score x: 85Mean μ: 70Std dev σ: 10

1
Subtract the mean
85 − 70= 15
2
Divide by the standard deviation
15 ÷ 10= 1.5
3
Convert to a percentile (standard normal CDF)
Φ(1.5)= 93.3193%

Result

Z-score: 1.5

A score of 85 is 1.5 standard deviations above the mean, placing it around the 93.3th percentile.

Z-scores for a mean of 100 and standard deviation of 15

A z-score is how many standard deviations a value sits from the mean. These use the classic IQ scale, but the arithmetic is the same for any distribution.

Value	Z-score	Meaning
70	-2.00	2 SD below the mean
80	-1.33	1.33 SD below
85	-1.00	1 SD below
100	0.00	Exactly at the mean
115	1.00	1 SD above
120	1.33	1.33 SD above
130	2.00	2 SD above
145	3.00	3 SD above

Z-scores make values from different scales directly comparable: a z of 1.5 on a test and a z of 1.5 on a height measurement describe the same relative standing. In a normal distribution roughly 68% of values fall within 1 SD, 95% within 2 and 99.7% within 3, so a z of 3 is genuinely rare at about 1 in 740 above that point. The z-score itself makes no assumption of normality - it is just a rescaling - but converting it to a percentile does, which is what the normal distribution calculator handles.

How many standard deviations away?

A z-score converts a raw value into 'number of standard deviations from the mean' — a z of +2 means the value sits two standard deviations above average, regardless of the original units.

Why standardising helps

Z-scores let you compare values from entirely different scales — a test score and a height measurement — on the same standardized footing, and directly translate to a percentile ranking.

Frequently asked questions

I scored 85 on a test where the class mean was 72 and SD was 8. How good is that?

z = (85 − 72) / 8 = 1.625. That puts you about 1.6 standard deviations above average, which corresponds to roughly the 95th percentile — better than about 95% of the class. Use the normal distribution calculator to find the exact percentile.

What is a 'good' z-score?

It depends on context. z = 0 means exactly average. z = +1 is better than ~84% of the group. z = +2 is better than ~97.7%. In quality control, a z-score beyond ±3 (the '3-sigma rule') signals something unusual — only 0.3% of data falls that far from the mean.

How is a z-score different from a percentile?

A z-score measures distance from the mean in standard deviations (can be negative). A percentile tells you what percentage of values fall below yours (always 0–100). They're related: z = +1 ≈ 84th percentile, z = 0 = 50th percentile, z = −1 ≈ 16th percentile.

Can I compare scores from different tests using z-scores?

Yes — that's their main purpose. If you scored z = 1.5 on a math test and z = 0.8 on an English test, you performed relatively better in math, even if the raw scores or scales are completely different. Z-scores remove the units and put everything on the same scale.