Z-Score Calculator

A z-score (standard score) measures how many standard deviations a value is from the mean. z = (x - mean) / standard_deviation. A z-score of 0 means exactly at the mean. z = +1 means 1 standard deviation above mean. z = -2 means 2 standard deviations below mean.

z = 1.65 corresponds to approximately the 95th percentile. Common z-score to percentile conversions: z=0 → 50th, z=1 → 84th, z=1.28 → 90th, z=1.65 → 95th, z=2 → 97.7th, z=2.33 → 99th, z=3 → 99.87th.

For normal distributions: 68% of data falls within 1 standard deviation of mean; 95% within 2 standard deviations; 99.7% within 3 standard deviations. A z-score beyond ±3 is very unusual — less than 0.3% of data in a normal distribution.

Z-scores standardize data across different scales. Example: a student scored 85 on math (mean=75, SD=8) and 76 on English (mean=70, SD=5). Math z = 1.25, English z = 1.2. Despite lower absolute English score, performance is very similar relative to peers.

Both measure standard deviations from the mean. Z-score: used when population mean and standard deviation are known. T-score: used when estimating from a sample (more common in practice). T-distributions have heavier tails than normal and adjust for small samples. For large samples (n > 30), t and z are nearly identical.