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z-score

A z-score (aka, a standard score) indicates how many standard deviations an element is from the mean. A z-score can be calculated from the following formula.

z = (X - μ) / σ

where z is the z-score, X is the value of the element, μ is the population mean, and σ is the standard deviation.

Here is how to interpret z-scores.

• A z-score less than 0 represents an element less than the mean.
• A z-score greater than 0 represents an element greater than the mean.
• A z-score equal to 0 represents an element equal to the mean.
• A z-score equal to 1 represents an element that is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean; etc.
• A z-score equal to -1 represents an element that is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean; etc.
• If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; and about 99% have a z-score between -3 and 3.

Here is another way to think about z-scores. A z-score is the normal random variable of a standard normal distribution .

See also:

Statistics Tutorial: Normal Distribution | AP Statistics Tutorial: Measures of Position | Normal Calculator