Z-Score
A z-score measures how many standard deviations a data value is from the mean, allowing comparison across different distributions.
Formula
z = \dfrac{x - \text{mean}}{\text{standard deviation}}
Definition
A z-score tells you how far a value is from the average, measured in standard deviations: a z-score of 0 means exactly average, positive z-scores are above average, negative ones are below. Formally, $z = (x-\mu)/\sigma$, where $\mu$ is the population mean and $\sigma$ is the population standard deviation (use $\bar{x}$ and $s$ for sample data); z-scores standardize data to have mean $0$ and standard deviation $1$, enabling comparison across different scales. When $X \sim N(\mu, \sigma^2)$, $z \sim N(0,1)$; for sample means, the z-score $z = (\bar{x}-\mu)/(\sigma/\sqrt{n})$ follows $N(0,1)$ exactly for normal populations and approximately by CLT for large $n$, forming the basis for z-tests and normal confidence intervals.
Example
On a test with mean $75$ and standard deviation $10$, a score of $90$ has $z = (90-75)/10 = 1.5$ and a score of $65$ has $z = (65-75)/10 = -1$, so the score of $90$ is $1.5$ standard deviations above average. Maria scored $82$ on a history test (mean $70$, SD $8$, $z=1.5$) and $78$ on a science test (mean $65$, SD $6.5$, $z=2.0$); her science score is relatively better despite being lower in raw points. A $95\%$ confidence interval for $\mu$ uses $z^* = 1.96$: for a sample of $n=100$ with $\bar{x}=52$ and $\sigma=10$, $CI = 52 \pm 1.96(10/10) = (50.04, 53.96)$.
Key Insight
Z-scores let you compare values from completely different datasets: a z-score of $+2$ on a math test and a z-score of $+2$ on a running time both mean "unusually high for that group," and z-scores above $2$ or below $-2$ are considered unusual (about $5\%$ of the time in a normal distribution), while those above $3$ or below $-3$ are very rare (about $0.3\%$). The probability integral transform states that $F(X) \sim \text{Uniform}(0,1)$ for continuous $X$; for a normal $X$, this implies $\Phi(z) \sim \text{Uniform}(0,1)$ where $\Phi$ is the standard normal CDF, linking z-scores to percentiles.