Quantiles

Quantiles are essentially points taken at regular vertical intervals from the [cumulative distribution]? of a random variable. Put another way, a quantile is the value such that the probability that a random variable will be less is _x_%.

Special quantiles include the percentiles, deciles, quintiles, quartiles, the minimum, the maximum and the median. There are 100 percentiles, each corresponding to a quantile represented by an even percentage (such as 99%). Deciles are the 0, 10, 20, 30 ... 90 and 100th percentiles. Quintiles are the 0, 20, 40, 60, 80 and 100th percentiles, quartiles the 0, 25, 50, 75 and 100th percentiles and so on. The minimum and maximum are the 0th and 100th percentile respectively. The median is the 50th percentile. If a distribution is symmetrical, then the median is the mean, but this is not generally the case.

Quantiles are useful measures because they are less susceptible to long tailed distributions and outliers. For instance, with a random variable that has an exponential distribution that has a mean of m, 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.

Empirically, if the data you are analyzing is not actually distributed according to your assumed distribution, or if you have other potential sources for outliers that are far removed from the mean, then quantiles may be more useful descriptive statistics and means and other moment related statistics.

Closely related is the subject of [robust regression]? 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. Robust regression shares the ability to be relatively insensitive to large deviations in outlying observations.