Key takeaways
- A percentage describes your own answers. A percentile describes other people. Scoring 80 per cent correct and ranking at the 80th percentile are unrelated statements that happen to share a number.
- The US Bureau of Labor Statistics gives the cleanest definition of a percentile: it is a boundary, with a fixed share of the distribution on either side of it.
- A standard score says how many standard deviations from the mean a result sits. On a scale with a mean of 100 and a standard deviation of 15, a score of 115 is one standard deviation up, which lands at roughly the 84th percentile.
- Percentile ranks are not equally spaced. Five points near the centre of the scale moves you about 13 percentile points; the same five points at the top moves you about 1.3.
- Every score carries measurement error, so a competent report shows a confidence interval. A single number implies a precision the test does not have.
- Accuracy in our own app, IQ Test & Brain Training Games, is a percentage of the questions it asked you, not a percentile. Its listing calls the results of its IQ-style test estimates based on app activities and documents no comparison group, so do not read any figure from it as a rank among other people.
A percentage is about you. A percentile is about everybody else. A standard score is about the distance between you and the middle of everybody else. These three numbers answer three different questions, and confusing any two of them will mislead you about a test result — which happens constantly, because they are all reported as bare numbers between 0 and 150.
The cleanest way to keep them apart is to notice what each one needs in order to exist. A percentage needs only your answers. A percentile and a standard score both need a comparison group, and are meaningless without one.
A percentage describes your own work
Items correct divided by items attempted, times one hundred. That is the whole definition, and it is self-contained: nobody else has to have taken the test.
A percentage is also entirely dependent on how hard the items were. Ninety per cent on an easy set and ninety per cent on a punishing one are the same number describing very different performances. This is exactly why norm-referenced tests do not report one.
A percentile is a boundary in a distribution
The US Bureau of Labor Statistics, explaining its wage estimates, gives the definition worth memorising: a percentile is a boundary. Its example is the median — the 50th percentile — which it describes as the boundary between the highest paid 50 per cent and the lowest paid 50 per cent of workers in an occupation.
Transfer that to testing and the meaning is exact. Sitting at the 84th percentile means 84 per cent of the comparison group scored at or below your level, and about 16 per cent scored above it. It says nothing at all about how many questions you answered correctly.
Two things follow. A percentile is only as meaningful as the group it is computed against; a rank among self-selected volunteers on a web page is not a rank among the general population. And a percentile can change without your performance changing, if the comparison group changes.
A standard score measures distance from the mean
A standard score expresses a result as a distance from the comparison group's average, in standard deviations. The scale used by the major cognitive batteries has a mean of 100 and a standard deviation of 15.
Converting between a standard score and a percentile is a property of the normal curve. The NIST statistical handbook supplies the three anchors: 68.27 per cent of a normal distribution lies within one standard deviation of the mean, 95.45 per cent within two, and 99.73 per cent within three. Everything else is arithmetic on those.
Take the one-standard-deviation figure. If 68.27 per cent sits between 85 and 115, then 31.73 per cent sits outside that range, split evenly between the two tails — 15.865 per cent above 115. So 115 sits at the 84.1st percentile. Repeat with the two-standard-deviation figure: 4.55 per cent outside, 2.275 per cent above 130, so 130 sits at the 97.7th percentile.
The table below applies that conversion across the usual landmarks, and is the most useful single thing to keep from this article.
| Standard score | Distance from mean | Approximate percentile |
|---|---|---|
| 70 | −2 standard deviations | 2nd |
| 85 | −1 standard deviation | 16th |
| 100 | at the mean | 50th |
| 105 | +0.33 standard deviations | 63rd |
| 115 | +1 standard deviation | 84th |
| 120 | +1.33 standard deviations | 91st |
| 130 | +2 standard deviations | 98th |
| 145 | +3 standard deviations | 99.9th |
Percentiles are not evenly spaced, and this matters
Look at what happens between adjacent rows. Moving from 100 to 105 — a third of a standard deviation — carries you from the 50th percentile to about the 63rd, a jump of roughly 13 percentile points. Moving from 130 to 135, the same five points of standard score, carries you from about 97.7 to about 99.0: a jump of roughly 1.3.
The same five points is worth ten times as much rank near the middle as it is near the top, because that is where the people are. Percentile ranks are ordinal positions on a bunched distribution, not equal-interval measurements.
The practical rule is short: never subtract percentiles, never average them, and never treat a percentile gap as a quantity. Compare them as positions and convert to standard scores before doing arithmetic. The way this plays out when the positions are given descriptive labels is covered in IQ score ranges and bands.
The number has a range around it
None of these figures is exact, and a good report says so. ETS, in its plain-language guide to reliability, defines reliable scores as those that are consistent across different occasions of testing, across different editions of a test, and across different raters scoring the same responses. The shortfall from perfect consistency is measurement error, and the standard error of measurement summarises how much of it to expect.
The consequence for reading a report is concrete. A standard score reported as a single integer implies a precision the instrument does not claim. A confidence interval around it — which is what qualified administrators report — shows the range the true value plausibly occupies, and it is wide enough that a rank near a boundary should never be treated as settled.
It is also worth noting that reported categories are definitions rather than natural facts. The National Center for Education Statistics makes this explicit for its own reporting: NAEP results are presented as scale scores and achievement levels, and the guidance states directly that NAEP Proficient does not represent grade-level proficiency and cannot be equated with another assessment's standard. A label means what its publisher defined it to mean, and no more.
Where our app's percentage fits
IQ Test & Brain Training Games is our own app, built by Reign Creative, and it reports accuracy on every run: the share of that session's questions you answered correctly.
That is a percentage in the strict sense above. It is computed from your answers to the questions the app generated for you, at the difficulty you chose, and it is not compared with anybody. The app also has an IQ-style test, whose results its Google Play listing calls estimates based on app activities, not a professional assessment, and the listing documents no comparison group for them. So do not read the accuracy figure, or any figure from the test, as a standing among other people — for accuracy, that would be a category error.
Accuracy is still the single most useful number the app gives you, provided you compare like with like. Hold the mode constant and hold the difficulty constant — Division Dash on the same setting, or the same length of number-memory round — and a change in accuracy across sessions is informative about your practice. Change the mode or the difficulty and the comparison is worthless. This is practice feedback, not standardised psychological assessment, and the value comes from tracking yourself rather than a population; our guide to building a daily practice habit covers doing that without fooling yourself.
Three questions, three numbers
How much did I get right? That is a percentage. Where did I rank? That is a percentile, and only against a stated group. How far from the middle was I? That is a standard score, and it converts to a percentile through the curve.
Keep those attached to the numbers and most test reporting becomes legible. For the conversion machinery, see how an IQ score is calculated; for what the underlying battery was sampling, what IQ tests actually measure. The rest of the series sits in the IQ tests and reasoning hub and the broader education and brain training section.
Good to know
Frequently asked questions
What is the difference between a percentile and a percentage?
How do you convert a standard score into a percentile?
Why is the 90th percentile not twice as good as the 45th?
Is accuracy in a practice app a percentile?
Sources
- Occupational Employment and Wage Statistics: Frequently Asked Questions (opens in a new tab)U.S. Bureau of Labor Statistics, U.S. Department of Labor — accessed
- 6.5.1. What do we mean by "Normal" data? (NIST/SEMATECH e-Handbook of Statistical Methods) (opens in a new tab)National Institute of Standards and Technology — accessed
- Test Reliability — Basic Concepts (Research Memorandum RM-18-01) (opens in a new tab)Educational Testing Service (Samuel A. Livingston, 2018) — accessed
- Understanding Results — The Nation's Report Card (NAEP) (opens in a new tab)National Center for Education Statistics, U.S. Department of Education — accessed
About the author
Reign Creative LLC — The independent app studio that builds the apps covered on this site and publishes every article here. On Google Play it publishes as Reign Collective Apps.
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