What Is Standard Deviation? The Number Behind Every IQ Score

Every IQ score is built on one statistical idea: standard deviation. It's the single number that turns "you answered 34 questions correctly" into "your IQ is 122." If you understand standard deviation, the rest of how IQ scores work falls into place.

The idea, without the formula first

Imagine you measure the height of 10,000 adults. Most cluster fairly close to the average. Some are noticeably taller or shorter. A few are extreme outliers. If you plotted every height on a graph, you'd get the familiar bell shape — the normal distribution.

Standard deviation answers one question: on average, how far does a typical person sit from the mean? A small standard deviation means most people are bunched close together. A large one means scores are spread wide.

The formula, briefly

For a set of scores, standard deviation (σ) is calculated as:

1. Find the mean (average) of all scores.
2. Subtract the mean from each score, and square the result (this removes negative signs).
3. Average those squared differences — this is the variance.
4. Take the square root of the variance. That square root is the standard deviation.

σ = √( Σ(xᵢ − μ)² / n )

Where μ is the mean, xᵢ is each individual score, and n is the number of scores. Squaring and then square-rooting looks roundabout, but it's what makes the result usable: it penalizes large deviations more than small ones, and the square root brings the units back to the original scale (points, centimeters, seconds — whatever you started with) instead of leaving you with squared units.

Why IQ tests are built around it

Test publishers don't just count correct answers. They give the test to a large, representative norming sample, record the mean and standard deviation of that sample's raw scores, and then convert every future test-taker's raw score into a standard score using that reference curve.

Almost every modern IQ test (Wechsler scales, Stanford-Binet, most Mensa admission tests) is normed to:

• Mean = 100
• Standard deviation = 15

That choice is a convention, not a law of nature — some older tests used a standard deviation of 16 or 24. But 15 is now the dominant standard, which is why an IQ of 130 means roughly the same thing across most major tests: it's 2 standard deviations above the mean, no matter which test produced it.

From standard deviation to percentile

Once you know how many standard deviations a score sits from the mean — its z-score — the normal distribution tells you exactly what percentage of people should score at or below it. That conversion follows the 68–95–99.7 rule:

• ±1 SD (85–115) contains about 68% of people
• ±2 SD (70–130) contains about 95% of people
• ±3 SD (55–145) contains about 99.7% of people

Try it directly: move the marker below and watch the z-score and percentile update together.

Two tests, two spreads

Here's the practical reason this matters. Suppose two different tests both report "your raw score is 1.5 standard deviations above your test's mean." On a test normed to SD = 15, that's an IQ of 122.5. On an older test normed to SD = 24, the same relative standing would be reported as IQ 136. Same person, same relative performance, very different-looking number — because the reported score depends on the scale's standard deviation, not just on how well you did.

This is also why comparing your score across two different IQ tests, or against a headline number from a forum post, is unreliable unless you know both tests use the same standard deviation and were normed on comparable samples. See our companion article on how IQ scores are calculated for the full raw-score-to-IQ pipeline.

The takeaway

Standard deviation isn't a technicality bolted onto IQ testing — it is the mechanism that makes an IQ score interpretable at all. It defines the ruler (how many points equal "one step away from average"), and every other statement about an IQ score — "top 2%," "gifted range," "borderline" — is just a restatement of how many standard deviations that score sits from 100.