Z-scores in simple words
A z-score tells you how far a number is from the average, measured in standard deviations (steps of typical spread). A z-score of 0 is exactly average. A positive z-score is above average, and a negative one is below.
Example: a class test has an average of 70 and a standard deviation of 10. You scored 85. Subtract the average: 85 − 70 = 15. Divide by the standard deviation: 15 ÷ 10 = 1.5. Your z-score is 1.5, so you are one and a half steps above average.
Z-scores let you compare things measured in different ways. A 1.5 in maths and a 0.5 in spelling means you did better in maths compared with your class, even if the tests were marked differently.
For data shaped like a bell curve, the z-score also gives a percentile. A z-score of 1.5 is about the 93rd percentile: about 93 out of 100 people scored lower.
Standardizing a value
Use z = (x − μ) ÷ σ, where x is the observation, μ the population mean, and σ a positive standard deviation. A positive z is above the mean; a negative z is below it. To reverse the operation, x = μ + zσ.
A z-score can describe distance from a mean without assuming a normal distribution. However, this page’s percentile and tail probabilities use the standard normal distribution, so interpreting them requires that model to be appropriate.
Example: score 85, mean 70, standard deviation 10
z = (85 − 70) ÷ 10 = 1.5. Under a normal model, about 93.3193% of observations are below this score and 6.6807% are above it. The probability outside ±1.5 standard deviations is about 13.3614%.
Percentiles and tails
The percentile mode inverts the cumulative normal distribution: a 90th percentile corresponds to z ≈ 1.281552. With mean 70 and standard deviation 10, that is a raw value near 82.815516.
Use percentiles strictly between 0 and 100. The endpoints correspond to infinite z-scores in an unbounded normal model. Standard deviation must be finite and greater than zero. Rounded probabilities near zero or one do not imply impossibility or certainty.