CalcWhenZ-Score Calculator
A score of 85 in a class averaging 70 with a standard deviation of 10 has a z-score of 1.50 — the 93.32nd percentile, ahead of about 93% of the class.
How it works
Raw scores are meaningless without their crowd: 85 is superb in a class averaging 70 and ordinary in one averaging 84. Enter the value, the mean and the standard deviation, and get the z-score — how many standard deviations above or below the mean the value sits — plus its percentile, the share of a normal distribution falling below it.
The panel also answers the two questions people actually ask next: what share scored above, and whether the value falls inside the familiar one- and two-standard-deviation bands that hold about 68% and 95% of a normal distribution.
The method
z = (x − mean) ÷ standard deviation. The percentile is the standard normal cumulative distribution at z, evaluated with the Abramowitz–Stegun 26.2.17 approximation (accurate to about 4 decimal places — plenty for interpretation; the panel's percentiles are quoted to 2).
Frequently asked questions
What does a z-score of 0 mean?
Exactly average — the value equals the mean. Positive z is above average, negative below; most values in real data land between −2 and +2.
What percentile is a z-score of 1.96?
About the 97.5th — 1.96 is the famous boundary of the middle 95% of a normal distribution, which is why it appears in every significance test.
Does the percentile assume my data is normal?
Yes — the percentile conversion reads z against the normal curve. Exam scores and measurements often approximate it; incomes and city sizes famously do not, and there the z-score still describes distance from the mean but the percentile is only indicative.