Standard score converter

What the converter does

Test reports for students, applicants and patients rarely use the same scale twice. One school sends home a T-score, another a stanine, a state test an NCE, a clinic an IQ-style deviation score. They all describe the same thing, how far a result sits from the average of a reference group, so this page converts a score from any of them into all the others. Pick the scale you have, enter the number and press Convert; the answer panel leads with the z-score because every other scale is a straight-line transformation of it.

What is a standard score, and why do tests have so many of them?

A standard score re-expresses a raw result as a distance from the average, measured in standard deviations. That distance is the z-score: z = (X − mean) ÷ SD. A z-score of 0 is average, +1 is one standard deviation above average, and −2 is two below. The other scales are the same number dressed up so that nobody has to read a negative value to a parent or a student: a T-score uses a mean of 50 and an SD of 10, an IQ-style deviation score uses a mean of 100 and an SD of 15, and an NCE uses a mean of 50 and an SD of 21.06. Because each is a straight-line transformation of z, you can move between them freely: T = 50 + 10 × z, IQ-style = 100 + 15 × z, NCE = 50 + 21.06 × z. Stanine (mean 5, SD 2) and sten (mean 5.5, SD 2) work the same way but are coarser, reporting only a whole-number band.

Worked example: a T-score of 65 and a raw score of 78

A T-score of 65 is 15 points above the mean of 50, which is 1.5 SDs, so z = 1.50. On a normal curve a z of 1.5 has a percentile rank of 93.3, which means the student scored higher than about 93 percent of the norm group. The same performance is an IQ-style score of 122.5 (SD 15), an NCE of 81.6, stanine 8 and sten 9. Now start from a raw score instead: 78 on a test with a mean of 70 and an SD of 4 gives z = (78 − 70) ÷ 4 = 2.00, a T-score of 70, an IQ-style score of 130 and a percentile rank of 97.7. Notice that a raw score is only meaningful against the mean and SD of the group it was normed on; ask the test publisher for them.

How are stanines and stens assigned?

Both split the normal curve into bands half a standard deviation wide. A stanine (short for “standard nine”) has nine bands whose middle one, stanine 5, covers z from −0.25 to +0.25; the cut points are at z = ±0.25, ±0.75, ±1.25 and ±1.75, and the two end bands are open-ended. In a normal population the bands hold roughly 4, 7, 12, 17, 20, 17, 12, 7, 4 percent of people, which is why the cumulative cut points are quoted as the 4th, 11th, 23rd, 40th, 60th, 77th, 89th and 96th percentiles. Stanine 2, for example, runs from z = -1.75 to -1.25. A sten (“standard ten”) has ten bands with cut points at every half SD from −2 to +2, so sten 5 is z from −0.5 to 0 and sten 6 is 0 to +0.5; its mean is 5.5, so unlike a stanine it has no middle score. This converter assigns bands from the z-score; some test publishers instead assign stanines from their own percentile tables, and those can differ by one band near a boundary.

Why does an NCE not equal the percentile rank?

Normal curve equivalents were designed to look like percentile ranks while keeping the equal-interval property of a z-score, so they can be averaged. The scale’s SD of 21.06 is chosen so that NCE 1, 50 and 99 coincide with percentile ranks 1, 50 and 99: the 99th percentile lies 2.3263 SDs above the mean, and 49 points ÷ 2.3263 = 21.06. Between those anchors the two scales disagree. A z of 1.5 gives NCE 81.6 but percentile 93.3. At the first percentile the formula gives 1.01 rather than exactly 1, because 21.06 is rounded. Percentile ranks are not equal-interval: the gap between the 50th and 60th percentile is much smaller in score terms than the gap between the 90th and 99th, so never average percentile ranks, average NCEs or z-scores and convert at the end.

Which IQ standard deviation should I use, 15 or 16?

Use the one printed in the test’s manual. Modern Wechsler scales and current versions of the Stanford–Binet report deviation scores with a mean of 100 and an SD of 15. Secondary sources describe the 1960 Stanford–Binet as using an SD of 16, and some older reports follow that convention, so the same percentile corresponds to a score a point or two higher on the SD-16 scale: at z = 1, 115 versus 116. The calculator shows both so you can match whatever report you are reading. This page is a score converter, not a diagnostic tool; classification of a child or adult should come from a qualified examiner using the full test.

What are the limits of these conversions?

Three limits matter. First, every percentile here assumes a normal curve; real score distributions are often skewed or have ceiling effects, and a norm-table percentile can differ from the normal-curve value, especially in the tails. Second, a z-score beyond about ±3 is based on very few people in any norm sample, so treat extreme values as “very high” or “very low”, not as precise ranks. Third, scores from different tests are only comparable if the groups they were normed on are comparable: a T-score of 60 against a national sample and a T-score of 60 against a gifted-program sample describe different performances. Rounding adds noise too, since a one-point change in a T-score is a tenth of an SD.

For general information only — not a psychological or educational assessment. Interpretation of an individual’s results belongs with a qualified professional and the test’s own manual. Sources: Wikipedia articles “Stanine” (band shares and z cut points), “Sten scores” (mean 5.5, SD 2; band shares), “Normal curve equivalent” (50 + 21.063 × z; cites C. A. Mertler, 2002, ERIC ED470589), “Standard score” (T-score, mean 50 and SD 10) and “Intelligence quotient” (mean 100, SD 15); the original test manuals were not opened. Related: z-score calculator, percentile calculator and standard deviation calculator. How we check our calculators.