Mathématiques

Standard Deviation

Roughly the typical distance from the mean. Two data sets sharing a mean of 50 compared, plus exactly where GCSE maths and the Digital SAT each stop with it.

Standard deviation

Standard deviation is a measure of spread: roughly, how far a typical value sits from the mean of its data set.

Aussi appelé: SD
Où les élèves le rencontrent: Not on the GCSE or IGCSE mathematics specifications we teach; it appears in GCSE Statistics and at A level, and on the Digital SAT as a comparison only.

La réponse en bref
Standard deviation measures spread: it is roughly the typical distance between a value and the mean. Two sets can share a mean of 50 and look nothing alike — 48, 49, 50, 51, 52 has a standard deviation of about 1.4, while 30, 40, 50, 60, 70 has one of about 14.1.

Un exemple

48, 49, 50, 51, 52 → SD ≈ 1.41; 30, 40, 50, 60, 70 → SD ≈ 14.14 (both have mean 50)

For the first set, the distances from the mean are −2, −1, 0, 1 and 2. Squaring gives 4, 1, 0, 1 and 4, which total 10; dividing by 5 gives 2, and the square root of 2 is about 1.41. The second set has distances ten times as large, so its squares are a hundred times larger and its standard deviation is ten times bigger, about 14.14. These are population standard deviations, dividing by n; the sample version divides by n − 1 and gives 1.58 and 15.81 for the same two sets.

Typical distance from the mean

Start with the obvious idea: to measure spread, find how far each value is from the mean and average those distances. The trouble is that the distances above the mean and below it cancel out exactly — they always total zero, for every data set, which makes the average distance useless as a measure.

Squaring solves that, since a squared distance is positive whichever side of the mean it came from. Averaging the squares gives the variance, but the variance is in the wrong units: square the distances between marks and you have marks squared, which nothing intuitive can be said about. Taking the square root at the end brings the answer back to marks, minutes or centimetres, and that is the standard deviation.

So the three steps of the formula are not arbitrary. Squares exist to stop the cancelling, the mean of the squares exists to average, and the root exists to undo the squaring. What comes out is close to the typical distance from the mean, weighted so that values far out count for more.

Same mean, different story

This is what standard deviation is for. Two classes averaging 50 marks are not the same class: one where every student scored between 48 and 52 needs different teaching from one spread from 30 to 70, and the mean alone cannot tell you which you are looking at.

The second set here has exactly ten times the spread of the first, and its standard deviation is exactly ten times larger. That proportionality is worth noticing — standard deviation scales with the data, so doubling every value doubles it, and adding 10 to every value leaves it unchanged, because shifting the whole set moves the mean along with it.

Where each exam stops

Standard deviation is not on the GCSE or IGCSE mathematics specifications we teach — not Edexcel 1MA1, not AQA 8300, not Cambridge OCR J560, not Cambridge International 0580. Spread at that level is handled by the range and the interquartile range. Students meet standard deviation in the separate GCSE Statistics qualification, or at A level.

The Digital SAT includes it, but only in one direction. Questions ask which of two data sets has the larger standard deviation, or how removing a value would change it — never to calculate one. The skill being tested is reading how tightly the values cluster around the centre, so an SAT student needs the concept and the comparison, and can leave the formula alone.

Questions fréquentes

Do I need standard deviation for GCSE maths?

No. The GCSE and IGCSE mathematics specifications we teach cover the range and the interquartile range as measures of spread and stop there. If your school also enters you for GCSE Statistics, that is a separate qualification and standard deviation is part of it, along with the formula and the calculation.

How is standard deviation different from the interquartile range?

Both measure spread, but differently. Standard deviation uses every value and is therefore pulled by extreme ones; the interquartile range discards the top and bottom quarters and is not. For roughly symmetric data with no unusual values, standard deviation says more; for skewed data or data with an outlier, the interquartile range is the safer summary.

Which do I divide by, n or n − 1?

Divide by n when the data is the whole population you care about — all 30 students in the class. Divide by n − 1 when the data is a sample being used to estimate a wider population's spread, because dividing by n underestimates it slightly. Calculators offer both, usually labelled σ and s.

What does a standard deviation of zero mean?

That every value in the set is identical. Each distance from the mean is zero, so every square is zero and so is the root. It cannot be negative, since it is the square root of an average of squares — a negative answer means an arithmetic mistake somewhere in the working.

Sources

Edexcel GCSE Mathematics (2015) specification — Pearson
The Math Section — SAT Suite of Assessments — College Board

Dernière mise à jour 21 août 2026