🎲 Z-Score

Z-Score#

How many standard deviations a value lies from the mean.

Important

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What it is#

A z-score (or standard score) says how many standard deviations a value sits from the mean of its distribution. It re-expresses raw numbers on a common, unit-free scale so values from different datasets can be compared directly.

The formula#

For a population,

\[z = \frac{x - \mu}{\sigma},\]

with raw value \(x\), mean \(\mu\) and standard deviation \(\sigma\). From a sample, use \(z = (x - \bar{x})/s\).

Reading it#

\(z = 0\) is exactly at the mean; \(z = +1\) is one SD above; \(z = -2\) two SD below. Large magnitudes (\(|z| > 3\)) flag likely outliers.

Examples#

An exam score of 85 with mean 70 and SD 10 gives \(z = (85 - 70)/10 = 1.5\) — 1.5 SD above average. A height of 150 cm with mean 170 and SD 8 gives \(z = (150 - 170)/8 = -2.5\).

Why it’s useful#

Three things at once: standardisation (compare maths and English scores on different scales), probability (in the standard normal — mean 0, SD 1 — a z maps to a tail area, e.g. \(z = 1.96\) bounds the central 95%), and test statistics (z-, t- and χ²-tests all compare an observed value to its expected spread in z-like units). A z-score standardises one data point; a z-test uses that machinery to test a hypothesis about a mean or proportion.

Theme: Probability & Statistics Foundations · All terminology

Hint

Mind map — connected ideas

Standard Error (SE) · Critical Value · Sample Standard Deviation · True Mean (Population Mean) · Two-Proportion Z-Test · Statistical Significance

Hint

More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion