How to use the standard score converter
- Choose the scale your score is on under Score is on this scale: a raw score, a z-score, a T-score, a standard score with mean 100, a scaled score with mean 10, a normal curve equivalent or a percentile rank.
- Type the Score. For a raw score also enter the mean and standard deviation of the group it is compared with, as printed in the test report or computed from your own data.
- Read the headline z-score first, then the stats list for the same position on every other scale, including the percentile rank and the stanine.
- Use Copy or CSV to keep the table of scales with your notes.
Formula
z = (score - mean) / standard deviation
T = 50 + 10 z standard score = 100 + 15 z scaled score = 10 + 3 z NCE = 50 + 21.06 z
percentile rank = 100 x (area under the normal curve below z)
Every common test scale is the z-score stretched to a different mean and spread, so converting between them is only arithmetic. The percentile rank and the stanine depend on the normal curve, so they are exact only when the scores follow one.
A raw score of 115 on a test with mean 100 and SD 15
The z-score is (115 - 100) / 15 = 1.000. That makes the T-score 60, the standard score 115, the scaled score 13 and the normal curve equivalent 71.06. The percentile rank is 84.13%, so a score this high or higher is reached by about 16 people in 100 under a normal curve. It falls in stanine 7, the band that holds the 77th to 89th percentile.
Going the other way, a percentile rank of 97.5 has z = 1.960, a T-score of 69.60 and a stanine of 9. A T-score of 30 has z = -2.000, a percentile rank of 2.28% and a stanine of 1.
What the scales mean
A z-score counts standard deviations from the mean, so 0 is average, 1 is one spread above and -2 is two spreads below. A T-score moves the mean to 50 and the spread to 10 so there are no negatives. A standard score with mean 100 and SD 15 is the form many ability and achievement reports use, and a scaled score with mean 10 and SD 3 is common for subtests.
Percentile rank tells you the share of the comparison group at or below a score. A stanine ("standard nine") sorts scores into nine bands holding 4, 7, 12, 17, 20, 17, 12, 7 and 4 percent of a normal distribution, from 1 at the bottom to 9 at the top.
When a conversion can mislead
The conversions assume the scores are spread like a normal curve. Real score distributions can be lopsided or have a ceiling, and then the percentile in the test publisher's norm table can differ from the one computed here. When a report gives a percentile directly, that figure is the better one to quote.
Different publishers use different means and spreads for their scales. This page uses the common ones listed above, so check the scale in your own report. To work from a list of raw scores, use the Z-Score and Normal Distribution Calculator or the Descriptive Statistics Calculator to find the mean and spread first.
More statistics tools
To plan a sample, look at runs of luck or measure agreement, try the Stratified Sample Allocation Calculator, Streak Probability Calculator and Exponential Distribution Calculator.
Assumptions and limits
- The scores follow a normal curve, which is what turns a z-score into a percentile rank and a stanine.
- The scales have the means and standard deviations shown in the table: T 50 and 10, standard 100 and 15, scaled 10 and 3, NCE 50 and 21.06.
- For a raw score, the mean and standard deviation you enter belong to the group the score is compared with.
- This page converts numbers only. It does not diagnose, classify or place anyone, and a single score is never a full picture.
- The example values are only an illustration. Replace them with your own before relying on the result.
Frequently asked questions
What is the difference between a percentile and a percent correct?
A percent correct is the share of questions answered right. A percentile rank is the share of other people who scored at or below you. Someone can answer 70% of questions correctly and still sit at the 90th percentile if the test was hard.
Why does a T-score of 50 match a standard score of 100?
Both mark exactly the mean, a z-score of 0. Each scale only chooses a different number to call the mean and a different size for one standard deviation, so the same position gets different numbers.
What is a normal curve equivalent?
A normal curve equivalent uses a mean of 50 and a spread of 21.06 so that 1, 50 and 99 line up with the same percentile ranks. It is used in some education reports to compare scores on an equal-interval scale.
Can I convert a stanine back to a score?
Not exactly, because a stanine covers a band of scores. This page gives the stanine for a score but does not guess a score back from a band. Use the percentile or z-score for that.