Is IQ Linear or Logarithmic? The Scale Is Linear, the Rarity Is Not
IQ scores sit on a straight-line scale: 100 plus 15 points for each standard deviation. The curve people sense is rarity, which thins out fast above 130.
What you need to know
- Is IQ linear? Yes. A modern IQ score is linear: IQ = 100 + 15 x z, where z is how many standard deviations you sit from the average for your age. There is no logarithm or exponent in the formula.
- Rarity is what curves. IQ 115 is the 84.1st percentile (about 1 in 6 score higher), 130 the 97.7th (1 in 44), 145 the 99.87th (1 in 741) and 160 the 99.997th (about 1 in 31,600).
- The old ratio IQ of the 1916 Stanford-Binet (mental age / age x 100) was not logarithmic either. Wechsler's 1939 test replaced it with the deviation IQ because ratio scores failed for adults.
- One step inside a test is curved: raw subtest points become scaled scores along the shape of the norm sample. The final composite still sits on the straight 100-and-15 scale.
No. IQ is a linear scale. A modern IQ score is 100 plus 15 times your z-score, the number of standard deviations you sit above or below people your age. There is no logarithm or exponent in that formula. What is not linear is rarity: each 15-point step above 100 passes fewer and fewer people, which is why high scores feel exponential.
Below: the formula with real numbers, a table of how rare each score is, why the old ratio IQ was not logarithmic either, and the one step inside a test that really is curved.
Is IQ linear?
Yes. Every major test today, including the Wechsler scales and the Stanford-Binet, reports a deviation IQ. The test first finds where you sit among people your age, as a z-score. Then one straight-line formula turns that position into a score: IQ = 100 + 15 x z.
- z = 0 gives IQ 100, the average of the norm sample by definition
- z = 1 gives IQ 115, one standard deviation above
- 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 shrinks
A worked example: if you score 1.4 standard deviations above people your age, your IQ is 100 + 15 x 1.4 = 121. Our guide to the IQ z-score works through more cases.
Is IQ logarithmic or exponential?
Is IQ logarithmic or exponential? Neither. A logarithmic scale squeezes each step at the top, as the decibel scale does. An exponential one stretches them. The IQ formula does neither: 115 is exactly as far above 100 as 100 is above 85. So is IQ exponential at the top? No: the IQ scale adds the same 15 points for every step.
Some tests use 16 or 24 points per standard deviation instead of 15. The scale is still a straight line; only the slope changes, as our article on SD 15 versus SD 16 scales explains. The IQ score converter moves a score from one to the other.
Why does a high IQ feel exponential?
Because rarity is not linear. The bell curve is crowded in the middle and thin at the ends, so the same 15 points pass very different numbers of people.
- IQ 100: 50th percentile, 1 in 2 people score higher
- IQ 115: 84.1st percentile, about 1 in 6 score higher
- IQ 130: 97.7th percentile, about 1 in 44 score higher
- IQ 145: 99.87th percentile, about 1 in 741 score higher
- IQ 160: 99.997th percentile, about 1 in 31,600 score higher
Moving from 100 to 115 passes 34 of every 100 people. Moving from 145 to 160 passes about 13 of every 10,000. The points are equal; the crowd they pass is not.
The same holds for small gaps. Scores of 98 and 106 sit about 21 percentile points apart. Scores of 138 and 146 sit less than half a percentile point apart. Our percentile calculator shows this for any score, and the IQ bell curve page draws it.
The score is linear. The rarity behind it is not, and that is the part people react to.
Was the old ratio IQ logarithmic?
No. Before deviation scores, the 1916 Stanford-Binet reported a ratio IQ: mental age divided by real age, times 100. A 10-year-old who performed like a typical 12-year-old scored 120. That is a ratio, not a logarithm.
Ratio IQ had two real problems. Mental age stops rising in adulthood, so the formula fails for adults. And the spread of ratio scores changed from one age to the next. David Wechsler's 1939 adult test replaced it with the deviation IQ, and the Stanford-Binet followed in 1960. Our explainer on deviation IQ versus ratio IQ tells that story in full.
What does this mean for your own score?
Read a 15-point gap as one standard deviation, wherever it falls. Read rarity separately: near 100 a few points pass many people, while near 145 they pass very few. Every score also carries a margin of error of a few points.
Where would your own score land?
Want your own score on both scales, as points and as a percentile? Take the IQ Metrics test.
Find your IQ score now! →A full IQ Metrics test reports both the score and its percentile, so you can see the linear number and the nonlinear rarity side by side.
Common questions
Is IQ logarithmic?
No. A deviation IQ is 100 + 15 x your z-score, a straight line. There is no logarithm in the formula, and 115 is exactly as far above 100 as 100 is above 85.
Is IQ exponential?
No. The points are equal steps. What grows fast is rarity: 130 is about 1 in 44, 145 about 1 in 741 and 160 about 1 in 31,600 on a scale with SD 15.
Is the IQ scale linear?
Yes. Modern tests such as the Wechsler scales and the Stanford-Binet report a deviation IQ, a linear transform of where you sit among people your age.
Why does a higher IQ feel exponential?
Because the bell curve thins at the ends. Going from 100 to 115 passes 34 of every 100 people; going from 145 to 160 passes about 13 of every 10,000.
What is the difference between ratio IQ and deviation IQ?
Ratio IQ (1916 Stanford-Binet) was mental age divided by age, times 100. Deviation IQ, introduced by Wechsler in 1939, places you on a bell curve of people your age with mean 100 and SD 15.
Sources for this story
- The Measurement of Intelligence (1916) — Lewis M. Terman, Houghton Mifflin
- The Measurement of Adult Intelligence (1939) — David Wechsler, Williams & Wilkins
- Standards for Educational and Psychological Testing (2014), on norms and standard scores — AERA, APA and NCME
- Normal-curve percentiles for mean 100 and SD 15, computed with Python statistics.NormalDist — IQ Metrics calculation
Corrections: spotted an error? Email corrections@iqmetrics.org and we will update this story and note the change here.
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Then find your own number.
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