The z-score formula
A z-score, or standard score, measures how far a value is from the mean in units of standard deviation:
z = (x − μ) ÷ σ
where x is your value, μ is the mean, and σ is the standard deviation. A score of 85 with a mean of 70 and a standard deviation of 10 gives z = 1.5 — one and a half standard deviations above average.
From z-score to percentile
Assuming a normal distribution, the percentile is the share of values that fall below your value. A z-score of 1.5 puts you at roughly the 93rd percentile — about 93% of values are lower. The "probability above" is simply the rest.
The empirical rule
- ±1 σ: about 68% of values fall within one standard deviation.
- ±2 σ: about 95% fall within two.
- ±3 σ: about 99.7% fall within three.
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from your value and divide by the standard deviation: z = (x − μ) ÷ σ. A value of 85 with a mean of 70 and a standard deviation of 10 gives z = (85 − 70) ÷ 10 = 1.5, meaning it sits 1.5 standard deviations above the mean.
What does a z-score tell you?
It tells you how many standard deviations a value is from the mean, and in which direction. A positive z-score is above the mean, a negative one is below, and zero is exactly average. Z-scores let you compare values from different data sets on the same scale.
How is the percentile found from a z-score?
The percentile is the area under the standard normal curve to the left of the z-score. A z-score of 1.5 corresponds to about the 93rd percentile, meaning roughly 93% of values fall below it. This calculator computes that probability for you.
What is a good or unusual z-score?
Most values in a normal distribution fall between −2 and +2 (about 95% of them). A z-score beyond ±2 is uncommon, and beyond ±3 is rare — only about 0.3% of values lie that far from the mean. What counts as "good" depends entirely on context.
Note: The percentile assumes a normal (bell-curve) distribution. It is computed with a high-accuracy approximation of the normal cumulative distribution.