The Z-Score

Introduces the z-score (standardized score) as a measure of how many standard deviations a value lies from the mean of its distribution. Covers computing z-scores from raw values, interpreting their sign and magnitude, recovering raw values from z-scores, and using z-scores to compare observations drawn from different distributions.

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Tutorial

Defining the Z-Score

The z-score (or standardized score) of a value $x$ measures how many standard deviations $x$ lies away from the mean. Given a distribution with mean $\mu$ and standard deviation $\sigma > 0$ , the z-score of $x$ is

z = \dfrac{x - \mu}{\sigma}.

A z-score of $z = 2$ means $x$ sits two standard deviations above the mean, while $z = -0.5$ means $x$ sits half a standard deviation below the mean.

For instance, take $x = 78$ in a distribution with $\mu = 70$ and $\sigma = 4$ . Substituting,

z = \dfrac{78 - 70}{4} = \dfrac{8}{4} = 2,

so $x = 78$ lies two standard deviations above the mean.