Calculate z-scores, percentiles, and probabilities from a normal distribution
How it works
A z-score restates a raw value as the number of standard deviations it sits from the mean. That puts measurements from completely different scales onto one common footing and links directly to probability under the normal curve.
z = (x − μ) / σ
xthe observed valueμpopulation meanσpopulation standard deviation
- 78 − 70 = 8
- 8 / 6 = 1.333
- Converting a z-score to a percentile requires the normal distribution, so the interpretation only holds if the data is roughly bell-shaped. The z-score itself is defined for any distribution.
- Common critical values: z = 1.645 for one-tailed 95%, z = 1.96 for two-tailed 95%, z = 2.576 for two-tailed 99%.
- With a sample standard deviation and a small sample, use the t-distribution instead. The two converge once n passes roughly 30.
- A rule of thumb flags |z| above 3 as an outlier, but in a sample of 10,000 normal values you should expect about 27 of them by chance alone.
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Frequently Asked Questions
A z-score measures how many standard deviations a value is above or below the mean. z = (x - μ) / σ. A z-score of 2 means the value is 2 SD above the mean.
About 95% of values in a normal distribution have z-scores between -2 and +2. Scores beyond ±3 are very unusual (less than 0.3% probability).
Look up the z-score in a standard normal table (or use a calculator). The resulting p-value tells you the probability of observing a value that extreme by chance alone.