The z-score (or standard score) tells you how many standard deviations a value lies above or below the mean: z = (x − μ) / σ. It puts values from any normal distribution onto one common scale, so a test score, a height and a lab reading can all be compared. The percentile is the share of the distribution that falls below the value, read from the standard normal curve.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the z formula and the standard normal CDF, recomputed in code.
The z-score formula
Subtract the mean (μ) from your value (x), then divide by the standard deviation (σ). A z of +1 means one standard deviation above the mean; −1 means one below; 0 is exactly at the mean. The percentile uses Φ, the standard normal cumulative distribution, which gives the proportion of values below z — so a larger z always means a higher percentile.
Worked examples
A score of 85 where the class mean is 70, σ = 10:
A score of 60 in the same class:
An IQ of 130 where μ = 100, σ = 15:
So 85 beats about 93% of the class, 60 beats about 16%, and an IQ of 130 is in the top ~2.3%. The sign shows the direction; the magnitude shows how unusual the value is.
Frequently Asked Questions
z = (x − μ) / σ. With μ = 70, σ = 10, a value of 85 gives z = 15/10 = 1.5.
How many standard deviations from the mean. z = 1.5 is 1.5 SD above; z = −1 is one below; z = 0 is the mean.
It's the normal-curve area left of z. z = 1.5 ≈ 93.3%, z = −1 ≈ 15.9%, z = 2 ≈ 97.7%.
Yes — it just means below the mean. z = −2 is two SD below, about the 2.3rd percentile.
The z-score is always valid; the percentile assumes roughly normal data. For skewed data it's approximate.