Z-Score

A z-score tells you how far a single data point sits from the average of its dataset, measured in standard deviations rather than raw units. It's a way of putting values from different scales, test scores, heights, stock returns, on the same comparable footing.

A z-score of 0 sits exactly at the mean. Positive values sit above it, negative values below. The further from zero, the more unusual the value is relative to the rest of the dataset. The Z-Score Calculator computes this directly from your data or from a value, mean, and standard deviation you supply.

Because it standardizes values, a z-score lets you compare a test score to a stock return in the same terms, both measured as "how many standard deviations from typical."

Z = (X − μ) / σ

Where X is the individual value, μ (mu) is the population mean, and σ (sigma) is the standard deviation.

A class's average test score is 75, with a standard deviation of 8. A student scores 91.

• Z = (91 − 75) / 8 = 16 / 8 = 2.0

That student scored two standard deviations above the class average, placing them near the top of the distribution.

• 68-95-99.7 rule applies under normal distributions. Roughly 68% of values fall within ±1 z-score, 95% within ±2, and 99.7% within ±3.
• Z-scores standardize different units. You can compare a z-score from exam results directly to one from income data, since both are expressed in standard deviation terms.
• A z-score near 0 is typical, not meaningless. It just tells you the value sits close to the average, which is often the expected outcome.
• Small sample sizes make z-scores less reliable. With very few data points, the mean and standard deviation themselves are shaky estimates, so treat the resulting z-score with caution.
• Used widely in finance for detecting anomalies. Unusual price movements or returns are often flagged using z-scores against historical volatility.