statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the percentile rank (PR) of a given score is the

percentage In mathematics, a percentage () is a number or ratio expressed as a fraction (mathematics), fraction of 100. It is often Denotation, denoted using the ''percent sign'' (%), although the abbreviations ''pct.'', ''pct'', and sometimes ''pc'' are ...

of scores in its

frequency distribution In statistics, the frequency or absolute frequency of an Event (probability theory), event i is the number n_i of times the observation has occurred/been recorded in an experiment or study. These frequencies are often depicted graphically or tabu ...

that are less than that score.

Formulation

Its mathematical formula is :

PR = \frac \times 100,

where ''CF''—the cumulative frequency—is the count of all scores less than or equal to the score of interest, ''F'' is the

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

for the score of interest, and ''N'' is the number of scores in the distribution. Alternatively, if ''CF'' is the count of all scores less than the score of interest, then :

PR = \frac \times 100.

Example

The figure illustrates the percentile rank computation and shows how the 0.5 × ''F'' term in the formula ensures that the percentile rank reflects a percentage of scores less than the specified score. For example, for the 10 scores shown in the figure, 60% of them are below a score of 4 (five less than 4 and half of the two equal to 4) and 95% are below 7 (nine less than 7 and half of the one equal to 7). Occasionally the percentile rank of a score is mistakenly defined as the percentage of scores lower than or equal to it, but that would require a different computation, one with the 0.5 × ''F'' term deleted. Typically percentile ranks are only computed for scores in the distribution but, as the figure illustrates, percentile ranks can also be computed for scores whose frequency is zero. For example, 90% of the scores are less than 6 (nine less than 6, none equal 6).

Usage

In educational measurement, a range of percentile ranks, often appearing on a score report, shows the range within which the test taker's "true" percentile rank probably occurs. The "true" value refers to the rank the test taker would obtain if there were no random errors involved in the testing process. Percentile ranks are commonly used to clarify the interpretation of scores on standardized tests. For the

test theory Test(s), testing, or TEST may refer to: * Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities Arts and entertainment * Test (2013 film), ''Test'' (2013 film), an American film * Test ...

, the percentile rank of a raw score is interpreted as the percentage of examinees in the norm group who scored below the score of interest.Crocker, L., & Algina, J. (1986). ''Introduction to classical and modern test theory''. New York: Harcourt Brace Jovanovich College Publishers.

Caveats

Percentile ranks are not on an equal-interval scale; that is, the difference between any two scores is not the same as between any other two scores whose difference in percentile ranks is the same. For example, is not the same distance as because of the bell-curve shape of the distribution. Some percentile ranks are closer to some than others. Percentile rank 30 is closer on the bell curve to 40 than it is to 20. If the distribution is

normally distributed In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...

, the percentile rank can be inferred from the

standard score In statistics, the standard score or ''z''-score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores ...

References

{{statistics Summary statistics

Formulation

Example

Usage

Caveats

See also

References