Quantile - Quantiles of A Population

Quantiles of A Population

For a population of discrete values, or for a continuous population density, the th -quantile is the data value where the cumulative distribution function crosses . That is, is a th -quantile for a variable if

(or equivalently, )

and

(or equivalently, ).

For a finite population of values indexed 1,..., from lowest to highest, the th -quantile of this population can be computed via the value of . If is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the th -quantile. On the other hand, if is an integer then any number from the data value at that index to the data value of the next can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see Estimating the quantiles).

If, instead of using integers and, the “-quantile” is based on a real number with, then replaces in the above formulae. Some software programs (including Microsoft Excel) regard the minimum and maximum as the 0th and 100th percentile, respectively; however, such terminology is an extension beyond traditional statistics definitions.