Z-Score Calculator

In short A z-score says how many standard deviations a value sits from the mean: z = (value − mean) ÷ standard deviation. A score of 130 where the mean is 100 and the standard deviation is 15 gives z = 2.0, which puts it at the 97.72nd percentile — higher than about 98 in 100 values. The sign carries the direction and the size carries the surprise, and because the units cancel, a z-score puts a height, an exam mark and a blood result on one scale. That is the entire point of it. Two things are worth knowing before trusting the percentile. First, the familiar 68–95–99.7 rule is rounded: one standard deviation covers 68.27% of a normal distribution, two covers 95.45% and three covers 99.73%. The value people actually want when they say "95%" is z = 1.96, not 2 — that gap is why confidence intervals use 1.96 and not the round number. Second, the percentage attached to a z-score is only true if the data really is normally distributed. The z-score itself is always computable; the percentile it implies is not always meaningful. For anything with a floor at zero and a long right tail — income, waiting times, response times — the normal percentages will be wrong, and a histogram settles it in seconds.