Is IQ Linear or Logarithmic? How the Scale Actually Works

Type "is IQ linear" into a search bar and autocomplete offers you "logarithmic," "exponential," and both, in roughly that order of popularity. The instinct makes sense: a jump from 100 to 130 feels like a bigger deal than a jump from 70 to 100, and unequal-feeling jumps are usually a sign of a curved scale. The instinct is also wrong about where the curve actually is.

The formula behind the number

Every modern IQ test — the Wechsler scales, the current Stanford-Binet, and effectively every serious instrument in use today — reports what is called a deviation IQ. A deviation IQ starts by finding a person's position on a standard normal distribution, expressed as a z-score: how many standard deviations above or below the average their raw performance falls. A standard deviation is simply a fixed unit of spread, chosen so that the scores of a large reference sample cluster around it in a known, predictable pattern. The final score is then: 100 plus 15 times that z-score.

That is a linear equation. There is no exponent, no logarithm, no curve of any kind in the formula itself. A z-score of exactly 1 gives 115. A z-score of exactly 2 gives 130. A z-score of exactly 3 gives 145. Each additional standard deviation adds exactly 15 more points, every time, with no acceleration and no flattening — which is the literal definition of a linear scale.

• z = 0 gives IQ 100, the mean of the reference sample by definition.
• z = 1 gives IQ 115, one standard deviation above the mean.
• z = 2 gives IQ 130, two standard deviations above.
• z = 3 gives IQ 145, three standard deviations above.
• Each step adds exactly 15 points. Nothing compounds, and nothing grows.

Where the "exponential" feeling actually comes from

The confusion is not baseless — it is just pointed at the wrong part of the system. What genuinely is nonlinear is percentile rank: the share of the reference population that a given score sits above. Because the bell curve is dense in the middle and thins out sharply toward the tails, an equal-sized move in IQ points buys you a wildly different amount of rarity depending on where you start.

Move from IQ 100 to IQ 115 — one standard deviation — and you jump from the 50th percentile to roughly the 84th: 34 percentile points for a 15-point move. Now make the identical 15-point move from 145 to 160, out at the far tail. You go from roughly the 99.87th percentile to roughly the 99.997th — a gain of about 0.13 percentile points for the same 15-point jump. Same distance on the score scale. A gap of over 250 times in what that distance is worth in rarity. That asymmetry is real, it is large, and it is almost certainly what people are actually noticing when they describe the scale as "feeling" exponential. Our IQ percentile calculator makes this concrete: plug in 115 and then 160 and watch how differently the percentile moves for the same 15-point step.

The same pattern shows up at gaps small enough to matter outside a research paper. Two children's school-administered scores of 98 and 106 — an 8-point gap, both close to average — sit about 21 percentile points apart. The identical 8-point gap between 138 and 146, both already rare scores, is worth less than half of one percentile point. A parent comparing those two pairs of numbers and expecting the second gap to "mean" roughly what the first one means is applying the score scale's logic to a place where only the percentile scale's logic actually applies.

The old formula that really was different — and still was not logarithmic

IQ testing did not start with deviation scoring. When Lewis Terman adapted the Binet-Simon test into the 1916 Stanford-Binet, the reported score was a ratio IQ: a child's mental age, as estimated from the test, divided by their chronological age, multiplied by 100. A ten-year-old performing like a typical twelve-year-old scored 120 under that formula. It is a different equation from deviation IQ, but it is still not a logarithm or an exponent — it is a ratio.

Ratio IQ had two real problems, and neither of them was curvature. Mental age effectively stops climbing in a meaningful way once someone reaches adulthood, so the ratio formula breaks down completely for adults — a 40-year-old does not have a sensible "mental age" the way a ten-year-old does. And the spread of scores was not constant across ages: a ratio IQ of 120 marked a different slice of the population at age eight than it did at age fourteen, which made scores hard to compare across an age range. David Wechsler's 1939 Wechsler-Bellevue scale fixed both problems by switching to deviation scoring — comparing each person only to others their own age, then converting that comparison to a fixed-spread standard score. The Stanford-Binet itself switched to deviation scoring in its 1960 revision, retiring the ratio formula it had used since 1916. One detail worth keeping in mind if you have compared old and new scores: some early deviation-scored tests used a standard deviation of 16 rather than 15, its own source of score confusion, covered in our piece on why one percentile has three different numbers depending on the standard deviation in use. A deeper look at exactly what changed in the formula itself is in our explainer on deviation IQ versus ratio IQ.

Why this matters when you are reading your own score

Get the mechanism right and two common mistakes disappear at once. First: do not assume a 15-point gap means the same thing wherever it falls. It means the same thing on the score scale — exactly one standard deviation, every time — but a wildly different thing in terms of rarity depending on whether you are near the middle of the distribution or out in a tail. Second: do not describe the score itself as compounding or accelerating. It does not. If something about an IQ comparison feels exponential, the feeling is accurate — you are just noticing the percentile underneath the score, not the score.

This is also why comparing two people's scores by simple subtraction works reasonably well near the middle of the distribution and gets less and less informative the further out you go. The distance in points stays constant by definition; the distance in rarity does not. Anyone reasoning carefully about a score difference needs both numbers, not just one.

IQ is not exponential, and it is not logarithmic. It is a linear scale wrapped around a distribution that itself is not linear in what it means — and almost every version of this question turns out to be a question about the second thing, mistakenly aimed at the first.

Common questions