Standard deviation in psychology

Why spread is the whole point

In many fields the standard deviation is a nuisance term — noise around the value you actually wanted. Psychology is different. Its subject matter is individual differences: why one person recalls nine words and another four, why anxiety scores on the same questionnaire run from floor to ceiling. The spread of a distribution is not error obscuring the signal; it very often is the signal.

That is why a mean reported on its own is close to useless in a psychology paper. Two groups with a mean reaction time of 420 ms could be a set of practically identical participants, or a mix of very fast and very slow ones whose average happens to land in the middle. The standard deviation is the number that tells those two apart, and reporting it is a matter of convention rather than choice.

A worked example with test scores

Ten participants sit a 100-point memory test and score 78, 85, 92, 81, 88, 74, 90, 83, 86 and 79. The scores sum to 836, so the mean is 836 ÷ 10 = 83.6. The next step is to see how far each participant sits from that mean, and to square the distance so that scores above and below do not cancel out.

The deviations sum to zero, as they always do — that is what the mean is. The squared deviations sum to 290.40. Because these ten people stand in for a wider population, divide by n − 1 = 9 rather than by 10:

s² = 290.40 ÷ 9 = 32.27 s = √32.27 = 5.68

So the sample has a mean of 83.60 and a standard deviation of 5.68 points. Read plainly: a typical participant scored within about six points of the average. The one score that was more than nine points away, participant 6 at 74, contributed nearly a third of the entire sum of squares on its own, which is squaring doing its job of weighting the far-out cases.

Had you divided by 10 instead, you would have got 5.39. That is the population standard deviation, and it is the wrong one here — see sample versus population for the reasoning. The calculator below is loaded with the same ten scores and shows every step.

Standardised scales built on the standard deviation

Once you know a distribution's mean and standard deviation, any raw score can be rewritten as a distance from the mean in standard-deviation units — a z-score. Participant 3 above, at 92, has z = (92 − 83.6) ÷ 5.68 = 1.48: about one and a half standard deviations above the group mean. Psychological tests go one step further and re-scale those z-scores onto a fixed mean and standard deviation, so that scores from different tests can be compared directly.

The IQ scale is the one everyone has met. Wechsler set the mean at 100 and the standard deviation at 15, and most modern tests follow. Earlier Stanford–Binet editions used 16, which is why an old score can be a point or two off a new one for the same underlying ability, and why a cut-off such as "IQ below 70" was never quite the same threshold on every instrument.

On a normal distribution, position in standard deviations translates straight into a percentile. One standard deviation above the mean — an IQ of 115, a T-score of 60 — is the 84th percentile. Two above (IQ 130, T-score 70) is roughly the 97.7th percentile, which is what a clinician means when they say a score is "in the top 2–3 per cent". Two below (IQ 70) is just above the 2nd percentile, and one below (IQ 85) is the 16th. The z-score calculator converts any raw score, and the percentile calculator turns the z into a percentile rank.

Reporting it in APA style

The convention in the seventh edition of the APA Publication Manual is to report the mean and the standard deviation together, with the statistical symbols in italics and a space either side of the equals sign. For the ten scores above:

Participants recalled a mean of 83.60 items (SD = 5.68, n = 10). — or — Memory scores were high (M = 83.60, SD = 5.68).

Three habits keep this clean. Give two decimal places unless the measurement itself is cruder. Never report a mean without its standard deviation — a reviewer will ask for it. And use the sample standard deviation, dividing by n − 1; that is what SPSS, R and jamovi return by default and what every reader will assume you used.

When you have more than two or three conditions, the numbers belong in a descriptive statistics table rather than the text. The layout is simple: one row per group or condition, columns for n, M and SD, with the symbols italicised in the headings. Some journals ask for a 95% confidence interval column beside the mean; it does not replace the standard deviation, because the two answer different questions.

Effect size: Cohen's d

A t-test tells you whether two means differ by more than chance would explain. It does not tell you whether the difference is large enough to care about, and with a big enough sample almost any difference clears the significance bar. Psychology's answer is Cohen's d: the difference between the means divided by the pooled standard deviation, so the gap is expressed in standard-deviation units rather than raw points.

d = (M₁ − M₂) ÷ s_pooled s_pooled = √[ ((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2) ]

Take the two conditions in the table. The spaced group has M = 83.60, SD = 5.68; the massed group, with scores of 76, 80, 71, 84, 79, 73, 82, 77, 75 and 81, has M = 77.80, SD = 4.13. With ten in each group the pooled variance is simply the average of the two variances:

s_pooled = √[ (9 × 32.27 + 9 × 17.07) ÷ 18 ] = √24.67 = 4.97 d = (83.60 − 77.80) ÷ 4.97 = 5.80 ÷ 4.97 = 1.17

The spaced group scored about 1.2 standard deviations higher than the massed group. Cohen (1988) offered 0.2, 0.5 and 0.8 as rough labels for small, medium and large effects, and those benchmarks are quoted in nearly every methods course. He also warned, in the same book, that they were arbitrary conventions to fall back on when a field had no better basis for judgement, and that a "small" effect in one literature can be a striking one in another. Report the number and let the reader judge it against the field, rather than the label alone.

The pooled SD calculator handles unequal group sizes, where the weighting matters, and the t-test calculator reports d alongside the test itself.

Standard deviation versus standard error

This is the confusion that costs the most marks. The standard deviation describes the participants: how widely their scores are spread. The standard error of the mean describes the estimate: how far the sample mean is likely to sit from the population mean. It equals the standard deviation divided by √n, so for the spaced group it is 5.68 ÷ √10 = 1.80.

Both are legitimate to report, but they are not interchangeable, and the standard error is always the smaller number. Error bars drawn from it look tight and persuasive; error bars drawn from the standard deviation look wide and honest about individual variation. A figure caption should always say which one it shows. If the point of the figure is to compare group means, the standard error or a confidence interval is the right choice; if the point is to show how variable people are, use the standard deviation. The standard error calculator gives both from the same data.

When the standard deviation is the wrong summary

The standard deviation assumes the mean is a sensible centre, and in three common psychological datasets it is not.

Reaction times are almost always skewed to the right: there is a hard floor of a couple of hundred milliseconds below which nobody can respond, but no ceiling on how long a lapse of attention can take. A handful of slow trials pulls the mean up and inflates the standard deviation far more than they shift the typical response. Reaction-time researchers routinely trim or transform the data before computing a mean and standard deviation, or report the median instead.

Single Likert items are ordinal — the distance from "agree" to "strongly agree" is not known to equal the distance from "neutral" to "agree" — so a mean and standard deviation on a single five-point item are of doubtful meaning. Summed scales of many items behave closely enough to interval data that the convention is to treat them as such, and reporting M and SD for a scale total is standard practice.

Very small samples give a standard deviation that is itself extremely uncertain. With five participants, one unusual score can double it. That does not make it wrong to report, but it does make it wrong to lean on.

In each case the median and the interquartile range are the robust alternatives, since neither is moved by a few extreme values. The IQR calculator reports both and flags outliers by the usual 1.5 × IQR rule, which is the first thing to run on a reaction-time column before deciding how to summarise it.