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The $$z$$ score for a particular individual is computed by subtracting the mean of the distribution ($$M$$) from that individual's raw score ($$X$$), and then dividing that difference by the standard deviation of the distribution ($$SD$$). The formula is expressed as: $$z = \frac{X - M}{SD}$$.

Statistical z Score Formula

Using the $$z$$ score formula, a raw score can be translated into standard deviation units. For example, in a distribution of intelligence quotient (IQ) scores with a mean of $$100$$ and a standard deviation of $$15$$, an IQ score of $$110$$ would have a $$z$$ score of $$\frac{110 - 100}{15} = +0.67$$. This indicates the score is approximately two-thirds of a standard deviation above the mean. Similarly, a raw score of $$85$$ would have a $$z$$ score of $$\frac{85 - 100}{15} = -1.00$$, placing it exactly one standard deviation below the mean.

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