Z-Score
Find how many standard deviations a value is from the mean.
The z-score calculator finds how many standard deviations a value sits from the mean: z-score = (value − mean) ÷ standard deviation, plus the corresponding cumulative probability and percentile.
Worked example
Using the numbers above:
- Value
- 115
- Mean
- 100
- Standard deviation
- 15
The formula
Variables
- Z-score
- Value
- Mean
- Standard deviation
What a z-score means
A z-score tells you how many standard deviations a value sits from the mean of its distribution. A z-score of 0 sits exactly at the mean; a z-score of +2 sits two standard deviations above it; a z-score of −1.5 sits one and a half standard deviations below it. Because it standardizes any value onto the same scale, a z-score lets you compare numbers that would otherwise be incomparable — an exam score, a height, a company's revenue growth — all in terms of "how unusual is this, relative to what's typical here."
Z-scores are the backbone of hypothesis testing: comparing a computed z-score against a critical value (like ±1.96 for a 95% two-tailed test) is how you decide whether an observed difference is statistically significant or just noise.
How to calculate it
Three inputs: the value, the mean, and the standard deviation of the distribution it belongs to.
- Subtract the mean from the value.
- Divide the result by the standard deviation.
That's the z-score. From there, the cumulative probability Φ(z) — the area under the standard normal curve to the left of z — converts the z-score into "what percentage of the distribution falls below this value," i.e. the percentile. This calculator computes Φ(z) with a standard numerical approximation (Abramowitz–Stegun), accurate to within 0.00002 percentage points.
- Use the sample standard deviation when your data is a sample of a larger population (the common case); use the population standard deviation only when your data genuinely is the entire population.
- The standard deviation must be positive — a value of 0 means there's no spread to measure a distance against.
A worked example
A student scores 115 on a test where the class mean is 100 and the standard deviation is 15. The z-score is (115 − 100) ÷ 15 = 1 — the student scored exactly one standard deviation above the mean. The cumulative probability Φ(1) ≈ 84.13%, meaning roughly 84% of the class scored at or below this student — putting them at the 84th percentile.
Limitations to know
The z-score and its percentile assume the underlying data is (at least approximately) normally distributed — for strongly skewed data, the percentile Φ(z) gives will be inaccurate, and a non-parametric approach is more appropriate. Z-scores also assume you know (or can reliably estimate) the true mean and standard deviation; with a very small sample, a t-score (which accounts for that extra uncertainty) is the more defensible choice — see the Data Analysis course below for when to switch.