Z-Score Calculator
Z-score, percentile rank, and plain reading from value, mean, and standard deviation.
What it does
A z-score puts any number on the universal ruler of standard deviations from the mean. Test scores, lab readings, and heights from different scales become comparable once each reads as so many deviations above or below its own average.
The tool above takes the value with the mean and standard deviation. It returns the z-score with the percentile rank under a normal curve and a plain reading beside them.
The default IQ of 120 against mean 100 and 15 deviation gives z = 1.333, about the 90.9th percentile and above average. An SAT of 600 against 500 with 100 deviation gives z = 1, near the 84.1st percentile by the same single division.
The definition follows the NIST Dataplot standard score reference with the Penn State statistics course teaching the same formula. Both pair it with the empirical rule: 68, 95, and 99.7% within 1, 2, and 3 deviations.
How to use
- Enter the single value to place.
- Enter the mean of its distribution.
- Enter the standard deviation, above 0.
- Read the z-score first with its sign.
- Read the percentile for the share below it.
Formula
- x
- The observation being placed.
- mean
- Center of the distribution.
- sd
- One standard deviation step on the same scale.
Percentiles assume a roughly normal shape. Skewed data keeps a valid z but the percentile share drifts. Sample statistics stand in for population values throughout this page.
Worked example
Example: IQ 120 with mean 100 and sd 15
Value 120, mean 100, standard deviation 15.
- Distance: 120 - 100 = 20.
- Divide: 20 / 15 = 1.333.
- Percentile: normal area below 1.333 is about 90.9%.
- Band: 1 to 2 reads Above average.
- Tool shows 1.333, 90.9th percentile, Above average.
Reference table
| Score | Z | Percentile |
|---|---|---|
| IQ 120 (mean 100, sd 15) | 1.333 | 90.9th percentile |
| SAT 600 (mean 500, sd 100) | 1 | 84.1th percentile |
| Mean itself | 0 | 50.0th percentile |
| IQ 85 (mean 100, sd 15) | -1 | 15.9th percentile |
Limits
- Percentiles assume a normal shape; skew bends them.
- One score never proves a trend; samples vary.
- Population versus sample deviation changes fine digits.
- Beyond 3 the tails need exact tables, not rules of thumb.
Common mistakes
- Dividing by variance instead of standard deviation.
- Reading z as a percent directly.
- Forcing percentiles onto skewed data.
- Comparing z values built from mismatched groups.
- Using 0 deviation, which has no defined score.
Try it yourself
Quick checks. Answers sit tucked under each question.
SAT 600 with mean 500 and sd 100?
(600-500)/100 = 1: 84.1th percentile, Above average.
85 with mean 100 and sd 15?
-1: 15.9th percentile, Below average.
Standard deviation of 0?
Rejected: enter a standard deviation above 0.
Questions
What is a z-score?
Distance from the mean counted in standard deviations. IQ 120 at mean 100, sd 15 is 1.333.
What does z = 1 mean?
One deviation above the mean, near the 84.1st percentile of a normal curve.
Can z be negative?
Yes. Below the mean reads negative: 85 at mean 100, sd 15 is -1.
How do percentiles follow?
Through the normal area below z. Rules of thumb: 68% within 1, 95% within 2, 99.7% within 3.
When does z mislead?
On skewed data the score stays valid but percentile shares drift from the normal table.
Sample or population values?
Either, stated plainly. This page accepts both and labels nothing beyond the three inputs.