Z-Scores & the Normal Distribution
A raw score on its own — an IQ of 130, a test score of 78 — doesn't tell you much. Is it high? Rare? It depends entirely on the rest of the data. A z-score fixes this by answering one clean question: how many standard deviations from the mean is this value?
The normal distribution
Lots of measurements — heights, reaction times, test scores, measurement errors — pile up in the same characteristic shape: a symmetric bell, tall in the middle, thinning out toward the edges. It's called the normal distribution, and it's fully described by just two numbers: its mean μ (where the peak sits) and its standard deviation σ (how wide the bell is).
The 68–95–99.7 rule: in any normal distribution, about 68% of values fall within 1 SD of the mean, about 95% within 2 SDs, and about 99.7% within 3 SDs. Drag the score below past the ±1σ, ±2σ, ±3σ marks and watch the percentile confirm it.
🎮 Z-Score Explorer
Example: IQ scores, which are built to have a mean of 100 and a standard deviation of 15. Drag the score and read off its z-score and percentile.
What a z-score actually is
The recipe is short: take how far the score is from the mean, then express that gap in standard-deviation units.
z = (x − μ) / σ
An IQ of 115 is (115 − 100) / 15 = +1.0 — exactly one SD above average. An IQ of 70 is (70 − 100) / 15 = −2.0 — two SDs below. The sign tells you which side of the mean you're on; the size tells you how unusual the value is. A z of 0 is dead average; a z beyond ±2 is genuinely rare (outside the middle 95%).
Why standardizing is so useful
- It makes different scales comparable. Scoring z = +1.5 on a vocabulary test and z = +0.5 on a math test means you were stronger, relative to peers, in vocabulary — even if the raw points looked similar.
- It unlocks percentiles. Because every normal distribution has the same shape once standardized, a z-score maps directly to "what fraction of people scored below this" — the percentile shown in the playground.
- It's the gateway to inference. The same standardizing move — distance from the expected value, measured in standard-error units — is exactly what test statistics and p-values are built on later.
Why it matters: z-scores turn "a score" into "a position." Once you can place any value on the normal curve and read its percentile, you've got the core skill behind confidence intervals, t-tests, and nearly everything that follows.