Raw Score Calculator
Standardised scores are a translation, and this calculator translates back: given a test’s mean and standard deviation, it converts a z-score or a t-score into the raw score it represents on the original scale, and reports the percentile the score beats. Test publishers, psychologists, clinicians and admissions offices all live in this conversion daily — a z of 1.5 means one thing on a scale averaged at 100 and quite another raw number on a scale averaged at 500 or fifty.
One Formula, Every Standardised Scale
All standard scores are the same animal at different zoos. A z-score states position in standard-deviation units from the mean, and the raw score is recovered by spending those units on the scale's own ruler: X = μ + zσ. T-scores merely re-dress z on a mean-50, SD-10 costume (T = 50 + 10z) to spare clinicians negative numbers and decimals; scaled scores, stanines, IQ metrics and SAT-style scales are further costumes on the same body. This calculator therefore needs only three inputs — the standard score, the scale's μ and σ — and it covers every costume by first stripping the score back to z, then applying the one formula. The percentile joins as a free companion, read off the normal curve: position in units converts immediately into proportion of the population below, which is the translation most readers actually wanted when they opened the report, the article, or the letter from the testing agency in the first place.
Worked Example: z = 1.5 on the IQ Ruler
The defaults are psychology's home scale: mean 100, standard deviation 15, and a z-score of 1.5. The raw score: 100 + 1.5 × 15 = 122.5 — call it an IQ of 122–123, sitting at the 93.32nd percentile: of every hundred people, about ninety-three score below. The same z on the SAT-style ruler (μ = 500, σ = 100 per section in its classic form) returns 500 + 150 = 650, and as a T-score the 1.5 appears as 65 before conversion, restating to the same z and the same raw answers. Three professions quote three numbers for one performance; this page is the interpreter's booth where they become one again.
The Reverse Direction and the Percentile's Fine Print
The formula inverts trivially (z = (X − μ)/σ), and most z-score tools on this hub run it that way; this page exists because the published numbers in reports and research papers are so often the standardised ones, leaving readers to reverse-engineer the raw performance. Two cautions guard the percentile. First, it presumes normality — the conversion is exact only when the scale's scores genuinely follow the bell; heavily skewed instruments (many clinical symptom scales) publish their own percentile tables instead, and theirs govern. Second, percentiles compress at the tails: between z = 2 and z = 3 the raw score climbs a full σ while the percentile inches from 97.7 to 99.87 — at the extremes, argue in z or raw units, where differences keep their size. The calculator prints both precisely so the argument can happen in the honest units.
Scales You Will Meet
A pocket directory of the rulers this page serves. IQ (Wechsler): μ 100, σ 15 — and its subtest scaled scores run μ 10, σ 3. Classic SAT sections: μ 500, σ 100 (modern versions re-centre, but the conversion logic is unchanged). T-scores in clinical use: μ 50, σ 10, where 70 (z = 2) is a common clinical threshold — a T of 70 on a μ 100/σ 15 scale converts to a raw 130 by this page's arithmetic. GRE (legacy): μ 500, σ 100. Stanines: a nine-band compression of z (bands centred at μ 5, σ 2) — too lossy to invert exactly, the cautionary tale of over-translation. Enter whichever pair defines your scale; the formula has read them all.
The Cautionary Cousins: Age and Grade Equivalents
Not every "standard" score inverts as cleanly as z and T, and the exceptions teach the rule. Age-equivalent and grade-equivalent scores ("reading at a 9.4-year level") report the age or grade whose median performance the examinee matched — ordinal labels, not interval measurements: the distance from grade 2.0 to 3.0 is not the distance from 6.0 to 7.0, they extrapolate wildly beyond the grades actually tested, and they cannot be fed to X = μ + zσ at all. Publishers print them because parents understand them; psychometricians tolerate them because the manuals carry the standard scores (this page's citizens) alongside. When a report offers both, convert and reason with the standard score, and let the equivalent serve as the illustration it was meant to be.
One Performance, Five Costumes: A Full Circuit
Close the loop with one performance carried through every scale this page serves, to prove the family is one animal. Take a score of 650 on a μ 500, σ 100 ruler. Its z: (650 − 500)/100 = 1.5. On the IQ ruler (100, 15): raw 122.5, the worked example's own. As a T-score: 50 + 10 × 1.5 = 65. On a subtest scaled score (μ 10, σ 3): 10 + 1.5 × 3 = 14.5. Percentile throughout: 93.32, invariant under every costume change, because the percentile is the position and the costumes are just rulers. Fluency in this circuit — recognising 65, 122.5, 14.5, 650 and "93rd percentile" as one fact in five dialects — is the professional literacy the calculator exists to give, and it is why the z-score, the naked position with no ruler at all, remains the family's lingua franca.
Norm Groups: The Ruler's Hidden Owner
Every scale's μ and σ are properties of a norm group — the reference population the test was standardised on — and the same raw performance converts differently against different groups: an IQ of 122.5 is read against the test's national standardisation sample, an admissions score against its year's cohort of test-takers. When norms are revised (as they periodically are, populations drifting upward), identical performances earn different standard scores under old and new rulers — the Flynn effect made a career of noticing. The calculator converts faithfully on whichever ruler you supply; the professional habit is to name the norm group beside the converted score, since a position without its population is only half a position.
Related Tools
More from the Statistics Calculators hub: the t-statistic Calculator and the Normal Approximation Calculator, companions in the standard-score family.
Frequently Asked Questions
How do I convert a z-score to a raw score?
Multiply the z by the scale's standard deviation and add the mean: X = μ + zσ. The worked example: z 1.5 with μ 100, σ 15 gives 122.5. The formula has no other moving parts — every standardised-to-raw conversion ever printed is this line with a costume change first (T-scores divide their distance from 50 by 10 to become z; this calculator does the costume change for you when you select T).
What raw score is a T-score of 65?
It depends on the target scale, which is this page's whole thesis: T 65 is z = (65 − 50)/10 = 1.5 wherever it lands — raw 122.5 on a μ 100/σ 15 scale, 650 on a μ 500/σ 100 scale, 57.5 on a μ 50/σ 5 scale. A T-score alone specifies a position, never a raw number, until a ruler is chosen; quote conversions with their scales attached or they are meaningless.
Why is the score 122.5 not an integer?
Because the conversion is exact arithmetic on a continuous ruler: raw-score scales that only award integers round at the end (an IQ of 122 or 123 depending on the publisher's convention), and the rounding belongs to the report, not the maths. This calculator keeps the decimal so chained conversions (raw to another scale, percentile lookups) don't accumulate rounding — round once, at the moment of publication, like a professional.
Does a z of 0 always mean the average raw score?
Yes by construction: z = 0 sits exactly at μ, so the raw score is the mean itself, percentile 50.00. Negative z converts symmetrically below (z = −1.5 on the IQ ruler: 77.5, percentile 6.68) — the bell's mirror symmetry, visible in this page's outputs, and a quick conscience check on any conversion you perform by hand.
Can I convert between two different tests with this?
Positionally, yes — and it is done daily (concordance tables between admissions tests are built exactly this way): convert test A's score to z using A's μ and σ, then convert z to test B's raw using B's rulers, as two passes through this calculator. The caveat is construct, not arithmetic: the conversion equates positions in two populations, which is legitimate for percentile-matched scales and increasingly speculative as the tests measure different things or norm on different groups.
What if my scale's mean and SD aren't published?
Then position-based conversion is impossible from the score alone, and the honest suppliers of the scale (its manual, its publisher's technical supplement) are the only source — norm tables exist precisely to carry this information. Beware folk values passed between forums and old course notes; a wrong σ silently poisons every conversion downstream of it, in either direction, without leaving any trace in the arithmetic itself. This calculator's inputs are deliberately the scale's documented parameters: garbage there converts with perfect confidence into wrong raw scores.