IQ 145: how rare is it — and can it even be measured?

Scores this high are extrapolated, not measured

A test is normed by giving it to a representative standardisation sample — typically around two to two and a half thousand people for a major clinical battery — and recording how that sample performed. Every reported score is a position within that observed distribution. The problem at 145 is arithmetic: if a score is about 1 in 740, then a sample of two thousand people would be expected to contain one or two people at that level.

So the norms up here are not observations. They are the normal curve, fitted to the middle of the distribution where the data is, and then extended outward on the assumption that it keeps the same shape. That assumption is untested precisely where it is being relied on, and there is no strong reason to believe real ability distributions stay perfectly normal in the extreme tails. A reported 145 is best read as "beyond the range this test can resolve", with the specific number carrying much more uncertainty than its two or three digits imply.

This is not a criticism of the tests. It is a limit built into what norming can do, and the responsible way to read a score out here is as a wide band rather than a point.

What "1 in 740" actually means here

The rarity figure attached to 145 — about 1 in 740 — is a prediction from the model, not a count of people. Nobody has tested enough of the population to verify it. When the denominator is larger than the standardisation sample, as it is here, the statement "about 1 in 740" is a statement about the curve rather than about anybody who was measured.

The practical reading is directional: 145 indicates ability far enough out that the test cannot place it precisely, and comparisons between two such scores are not informative. "One in 740" is a reasonable way to convey extreme rarity and a bad way to rank individuals.

The band starts at 145 because that is where measurement stops

145 is exactly three standard deviations above the mean, and it is also where classification tables stop subdividing: everything above it is grouped together as one open-ended category. Those two facts are connected. A score of 145 is around one in seven hundred and forty, so a standardisation sample of a couple of thousand people would be expected to contain roughly one person at this level — and none above it.

So the label does not stop being specific out of modesty. It stops because there is nothing left to be specific with: with no observations above this point, any further subdivision would be dividing up a fitted curve rather than a measured population. 145 marks the place where the scale's numbers stop describing people who were tested and start describing an extrapolation.

Past every conventional threshold

145 is well above the two thresholds people usually ask about. The gifted line sits at 130 — a figure adopted because it is a round two standard deviations above the mean and because Mensa's 98th-percentile rule needed a number written into it, not because anything changes in the distribution there. Tests normed with a 16-point standard deviation put the same percentile at 132. Both are administrative cut-offs, and 145 clears them by a wide margin.

The popular 140 "genius" figure is looser still: it survives from Lewis Terman's 1920s studies, published as Genetic Studies of Genius, and appears in no current classification system. Clearing these lines by 15 points is worth less than it sounds, because none of the lines mark real boundaries.

Why 141 and 149 are effectively the same score

Measurement error does not shrink as scores rise — it grows, because there are fewer items and fewer norming subjects supporting each point. A reported 145 on a short test is consistent with a true value anywhere across roughly 141 to 149, and probably wider than that out here.

Which means the difference between 141 and 149 is noise, not information. If you have two results in this range from different tests, the gap between them tells you about the tests, not about a change in ability. Any argument that depends on a few points at this level is an argument about rounding.

How the open-ended 145+ band came to exist

Classification tables used to keep subdividing above 145. Mid-century Wechsler and Stanford–Binet tables carried named categories well into the 160s, inherited from an era less cautious about extrapolation. Modern tables mostly stop, and group everything above about 145 into a single open category. That change is a deliberate retreat: as norming practice grew more rigorous about what a standardisation sample can actually support, publishing distinct labels for scores nobody in the sample achieved stopped being defensible. The open band is a record of the field becoming more careful, not of the scale running out of room.

The edge of the rule everyone has heard of

The empirical rule taught with every normal distribution — about 68% of values within one standard deviation, 95% within two, 99.7% within three — puts its last landmark exactly here. Three standard deviations above the mean on this scale is 145, so 145 sits at or just past the boundary of the interval containing 99.7% of people.

That is why summaries of the distribution tend to stop at this point: it is the last position the familiar rule has a number for. Beyond it the description has to switch from round percentages to tail probabilities, which is a change in the kind of statement being made as much as in its precision.

Extended norms and high-ceiling tests

Some publishers do provide extended norms that report scores above a test's standard ceiling. These are derived statistically — by modelling item difficulty and projecting the scale upward — rather than from people who actually scored there, so they extend the reach of the reporting scale without adding observations at the top. They are useful for distinguishing "very far out" from "extremely far out" in research and in gifted-programme placement, and they are not precise individual measurements.

Historically, purpose-built high-ceiling instruments were used for the same job; the older Stanford–Binet forms had more headroom than most modern batteries, which is one reason very high figures quoted from decades ago are not comparable with scores from tests in use now.

What the number is still good for

A reported 145 does carry information. It tells you a test ran out of headroom before it ran out of you, which is a real and unusual finding, and it is enough to justify a supervised assessment on a battery with more ceiling if there is a decision resting on it — a placement, an acceleration, a research inclusion criterion.

What it does not support is treating the specific figure as a rank. The honest summary of 145 is "far out in the tail, further than this instrument can resolve" — and the subtest profile, with its own confidence intervals, is more informative than the composite in every practical respect.

At exactly three standard deviations out, the strand figures behind a 145 carry confidence intervals wide enough that their ranking against one another is often not resolvable — the profile shape is readable in broad terms, the ordering within it usually is not.