An IQ Score Is a Range, Not a Number: The Margin of Error

An IQ result arrives as one number, printed once, with no hint that it could have come out differently. That presentation is the problem. Every reputable test manual reports a second figure next to the score that describes how much the score itself wobbles, and once you know that figure most of the arguments people have about IQ points stop being arguments about anything.

What the reported number actually is

A modern IQ score is a position, not a quantity. The test is given to a large reference sample, the results are arranged so that the average sits at 100, and the spread is fixed by a chosen standard deviation — a measure of how far apart people's results fall. A score of 115 on a standard-deviation-15 scale means "one standard deviation above the average of that reference sample", and nothing else. It is not a count of anything.

Because it is a position estimated from a finite set of questions on a particular morning, it inherits every source of noise in that process. The specific items sampled, whether the room was quiet, how well the person slept, whether they had sat something similar before, ordinary fluctuation in attention — all of it moves the number without moving the thing the number is supposed to represent.

The standard error of measurement

Test developers do not leave that noise undescribed. They quantify it as the standard error of measurement, usually shortened to SEM: an estimate of the typical distance between a person's reported score and the score they would average over many independent sittings. It is derived from the test's reliability and its standard deviation, and it is published in the technical manual for every score the test produces.

For a full-scale score on a well-constructed adult battery scored with a standard deviation of 15, that error is commonly in the region of 2 to 3 points. Turning it into a confidence interval is ordinary arithmetic — about 1.96 standard errors either side for a 95 per cent interval:

• A standard error of 2.5 points gives a 95 per cent interval of roughly 5 points either side.
• A reported 112 therefore describes a person whose true standing is plausibly anywhere from about 107 to about 117.
• A reported 100 describes a range of roughly 95 to 105 — which spans, on an SD-15 scale, something like the 37th to the 63rd percentile.
• The interval belongs to the score, not to the person. Repeating the test does not narrow it; it simply produces a second number inside the same band.

Why two different scores can mean the same thing

Set two results side by side. One person reports 104, another reports 111. The gap looks real — seven points, and one of them is above average by more. But each of those numbers carries a band of roughly five points either side, and the bands overlap across almost their whole width. There is no basis in the scores for saying the second person scored higher in any durable sense. The honest statement is that the two results are not distinguishable.

The same logic applies to one person tested twice. A rise from 104 to 111 between two sittings is comfortably inside what measurement error alone produces, before considering the practice effect — the well-documented tendency for a second sitting on a familiar format to come out higher regardless of any change in ability. Our piece on what a retest actually measures works through that separate problem in detail.

Where the interval gets wider

The full-scale score is the most reliable number a battery produces, because it pools the most items. Everything beneath it is shorter and therefore noisier, and the published standard errors reflect that:

• Index scores — the groupings for verbal comprehension, working memory, processing speed and so on — typically carry larger standard errors than the full-scale total, so their intervals are wider.
• Individual subtest scores are shorter still and are the least stable figures on the report. A subtest result quoted on its own carries very little information.
• Differences between two index scores inherit the error of both, so a gap has to be substantial before it means anything. Manuals publish tables of how large a difference must be to be treated as real.
• Short-form and abbreviated tests trade items for time, and their intervals are wider than the full battery they approximate.

None of that makes the scores useless. It makes them estimates with known precision, which is a great deal better than most numbers people quote about themselves. The failure is not in the test, it is in reporting one figure from it as though it were exact. The same question of build quality applies before any of this arithmetic is worth doing — an unnormed test has no meaningful error term at all, because there is no reference sample for the score to be an estimate of.

What to do with a score you already have

Three things, in order. Name the scale it was measured on, because a bare number is uninterpretable without its mean and standard deviation. Convert it to a percentile against that scale, which is the form that survives translation between tests. Then read it as a range rather than a point, and describe it that way to anyone who asks. Our IQ percentile calculator does the first two conversions, and treating the answer as a band rather than a figure is the whole discipline.

The habit generalises well beyond testing. Any measurement of a person made once, on one day, from a sample of their behaviour, is an estimate with a band around it. Test manuals are unusual mainly in publishing the band. When someone quotes a score without one — their own, a public figure's, a country's average — the missing interval is not a detail that was left out for brevity. It is the part that tells you whether the number can bear the weight being put on it.

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