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Quantiles are values taken at regular intervals from the inverse of the cumulative distribution function (CDF) of a random variable. Dividing ordered data into q essentially equal-sized data subsets is the motivation for q-quantiles; the quantiles are the data values marking the boundaries between consecutive subsets. Put another way, a k^\mathrm{th} q-quantile for a random variable is a value x such that the probability that the random variable will be less than x is at most k/q and the probability that the random variable will be greater than x is at most (q-k)/q=1-(k/q). There are q-1 of the q-quantiles, one for each integer k satisfying 0 < k < q. In some cases the value of a quantile may not be uniquely determined, as can be the case for the median of a uniform probability distribution on a set of even size.
Some q-quantiles have special names:
More generally, one can consider the quantile function for any distribution. This is defined for real variables between zero and one and is mathematically the inverse of the cumulative distribution function.
For a population of discrete values, or for a continuous population density, the kth q-quantile is the data value where the cumulative distribution function crosses k/q. That is, x is a kth q-quantile for a variable X if
and
For a finite population of N values indexed 1,...,N from lowest to highest, the kth q-quantile of this population can be computed via the value of I_p = N \frac{k}{q}. If I_p is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the kth q-quantile. On the other hand, if I_p 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 k and q, the “p-quantile” is based on a real number p with 0, then p replaces k/q 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.
, then p replaces k/q 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.
The following two examples use the Nearest Rank definition of quantile with rounding. For an explanation of this definition, see percentiles.
Consider an ordered population of 10 data values {3, 6, 7, 8, 8, 10, 13, 15, 16, 20}. What are the 4-quantiles (the "quartiles") of this dataset?
So the three 4-quantiles (the "quartiles") of the dataset {3, 6, 7, 8, 8, 10, 13, 15, 16, 20} are {7, 9, 15}.
Consider an ordered population of 11 data values {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20}. What are the 4-quantiles (the "quartiles") of this dataset?
So the three 4-quantiles (the "quartiles") of the dataset {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20} are {7, 9, 15}.
Standardized test results are commonly misinterpreted as a student scoring "in the 80th percentile," for example, as if the 80th percentile is an interval to score "in," which it is not; one can score "at" some percentile, or between two percentiles, but not "in" some percentile. Perhaps by this example it is meant that the student scores between the 80th and 81st percentiles, or "in" the group of students whose score placed them at the 80th percentile.
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 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 m is the median of a random variable X, then 2^m is the median of 2^X, unless an arbitrary choice has been made from a range of values to specify a particular quantile. (See quantile estimation, below, for examples of such interpolation.) Quantiles can also be used in cases where only ordinal data are available.
There are several methods for [3] SAS includes five sample quantile methods, SciPy and Maple both include eight,^{[4]}^{[5]} STATA includes two, and Microsoft Excel includes one.
In effect, the methods compute Q_{p}, the estimate for the kth q-quantile, where p = k / q, from a sample of size N by computing a real valued index h. When h is an integer, the hth smallest of the N values, x_{h}, is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from h, x_{⌊h⌋}, and x_{⌈h⌉}. (For notation, see floor and ceiling functions).
Estimate types include:
Note that R-3 and R-4 do not give h = (N + 1) / 2 when p = 1/2.
The standard error of a quantile estimate can in general be estimated via the bootstrap. The Maritz-Jarrett method can also be used.^{[6]}
Statistics, Regression analysis, National Institute of Standards and Technology, Survey methodology, Multivariate statistics
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