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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).
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
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the percentile rank (PR) of a given score is the percentage
In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also us ...
of scores in its frequency distribution
In statistics, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form.
Types
The cumula ...
that are less than that score. Its mathematical formula is
:
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. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
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
:
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).
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
In experimental physics, a test theory tells experimenters how to perform particular comparisons between specific theories or classes of theory.
Without a good reference test theory, these experiments can be difficult to construct. Different theor ...
, 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.
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, the percentile rank can be inferred from the standard score
In statistics, the standard 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 above the mean ...
.
See also
*Quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...
* Percentile
In statistics, a ''k''-th percentile (percentile score or centile) is a score ''below which'' a given percentage ''k'' of scores in its frequency distribution falls (exclusive definition) or a score ''at or below which'' a given percentage falls ...
References
{{statistics Summary statistics