What percentiles are IQ scores of 115, 130 and 145?

The calculation uses z = (score − 100) / 15 and the cumulative standard normal distribution, rounded to two decimal places. Mean 100 and SD 15 are common scale conventions. Actual test manuals determine the appropriate norms, score ranges and interpretation; a mathematical normal curve does not validate a new test. Neisser et al., 1996 Pearson, 2020

A percentile is not a percentage correct

A percentile concerns standing in a reference distribution. It does not say that a person answered that percentage of items correctly or possesses that percentage of all intelligence. Test-use guidance requires a justified reference group and evidence for the intended score interpretation. Without those, a percentile can look more precise than its basis permits. International Test Commission, 2001

What changes with age?

IQ is age-normed: the reference mean is 100 within each age group on the conventional scale. A child and an adult can receive the same age-adjusted score while answering different items or obtaining different raw totals. The normal-model arithmetic therefore does not describe an unadjusted lifelong performance curve. Neisser et al., 1996 Pearson, 2020

Norms also depend on test, population and period. Raven's review documents variation in matrix-test norms across settings and time. A score from one instrument should not be treated as though every test and historical norm table shared identical measurement properties. Raven, 2000

Why the extreme tail deserves extra care

Small norm samples contain little direct information about very rare scores. An attractive tail percentile can therefore depend on model assumptions as well as observations. Test-use guidance calls for evidence appropriate to the inference, and a score report's uncertainty is more informative than treating the last decimal place as exact. International Test Commission, 2001 Pearson, 2020

Does a matrix puzzle score define overall IQ?

Carpenter and colleagues analyse the reasoning processes in Raven matrices. That gives a theoretical account of an important task family, not a licence to convert any homemade puzzle total into a normed overall IQ. Work on the structure of cognitive performance also cautions against assuming every task measures one identical aspect of ability. Carpenter et al., 1990 Hampshire et al., 2012

What about retests and measurement error?

A meta-analysis finds systematic retest gains in cognitive ability tests. A repeated test result may therefore reflect familiarity as well as other differences. Reliability research adds that task scores and differences need measured stability before being treated as individual traits. Neither issue is repaired by displaying more percentile digits. Scharfen et al., 2018 Hedge et al., 2017

For an established assessment, use its own report, age norms and confidence interval. For an exploratory online task, keep the result at the level its validation supports. These mathematical examples explain a common scale; they do not justify telling a player they are smarter than a particular percentage of people. International Test Commission, 2001 Pearson, 2020

The useful distinction is between understanding how a percentile is calculated and having evidence that a particular score belongs on that curve. Both are necessary before making a personal comparison. Neisser et al., 1996 International Test Commission, 2001