Z-Score

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z-Score (Standard Score)

Definition

The z-score (also called a standard score) measures how many standard deviations away from the mean a data point is. It allows us to compare values from different normal distributions and understand the relative position of any data point within its distribution.

Formula

For Population Data:

z = \frac{x - \mu}{\sigma}

$x$ = individual value
$\mu$ = population mean
$\sigma$ = population standard deviation

For Sample Data:

z = \frac{x - \bar{x}}{s}

$\bar{x}$ = sample mean
$s$ = sample standard deviation

Example

For a dataset with $\text{mean} = 75$ and $\text{standard deviation} = 8$ , calculate the z-score for a value of 83:

\begin{align*} z &= \frac{x - \mu}{\sigma} \\ &= \frac{83 - 75}{8} \\ &= 1 \end{align*}

The value 83 is one standard deviation above the mean.

Common Z-Score Values

z = 0

: Value equals the mean

|z| = 1

: One standard deviation from mean (encompasses ~68% of data)

|z| = 2

: Two standard deviations from mean (encompasses ~95% of data)

|z| = 3

: Three standard deviations from mean (encompasses ~99.7% of data)

Limitations & Considerations

Assumes data follows a normal distribution
Standard deviation must be greater than zero
Sensitive to outliers in small samples
May not be meaningful for non-normal distributions

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