Editing Quantile (section)

== Discussion ==
Standardized test results are commonly reported as a student scoring "in the 80th percentile", for example. This uses an alternative meaning of the word percentile as the ''interval'' between (in this case) the 80th and the 81st scalar percentile.<ref>{{Cite journal |title=percentile |url=https://www.oxfordreference.com/view/10.1093/oi/authority.20110803100316401 |access-date=2020-08-17 |website=Oxford Reference |doi= |language=en }}</ref> This separate meaning of percentile is also used in peer-reviewed scientific research articles.<ref>{{Cite journal |last1=Kruger |first1=J. |last2=Dunning |first2=D. |date=December 1999 |title=Unskilled and unaware of it: how difficulties in recognizing one's own incompetence lead to inflated self-assessments |url=https://pubmed.ncbi.nlm.nih.gov/10626367/ |journal=Journal of Personality and Social Psychology |volume=77 |issue=6 |pages=1121–1134 |doi=10.1037/0022-3514.77.6.1121 |issn=0022-3514 |pmid=10626367|s2cid=2109278 }}</ref> The meaning used can be derived from its context.

If a distribution is symmetric, then the median is the mean (so long as the latter exists). But, in general, the median and the mean can differ. For instance, with a random variable that has an [[exponential distribution]], any particular sample of this random variable will have roughly a 63% chance of being less than the mean. This is because the exponential distribution has a long tail for positive values but is zero for negative numbers.

Quantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers. Empirically, if the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers that are far removed from the mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics.

Closely related is the subject of [[least absolute deviations]], a method of regression that is more robust to outliers than is least squares, in which the sum of the absolute value of the observed errors is used in place of the squared error. The connection is that the mean is the single estimate of a distribution that minimizes expected squared error while the median minimizes expected absolute error. [[Least absolute deviations]] shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of [[robust regression]] are available.

The quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if {{mvar|m}} is the median of a random variable {{mvar|X}}, then {{math|2<sup>''m''</sup>}} is the median of {{math|2<sup>''X''</sup>}}, unless an arbitrary choice has been made from a range of values to specify a particular quantile. (See quantile estimation, above, for examples of such interpolation.) Quantiles can also be used in cases where only [[Ordinal scale|ordinal]] data are available.