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What is a Quantile?

A quantile is a cut point, or line of division, that splits a probability distribution into continuous intervals with equal probabilities. Naturally, there is always one less quantile than the number of groups created (i.e. one quantile splits a distribution into two sections).

Q-quantiles are quantiles that cut a set into "Q" subsets of nearly equal sizes. There are various names for the specialized forms of quantiles. For example, a 2-quantile, or a quantile with 2 partitions, is also called the median, a 3-quantile is called a tercile, or tertile, and so on. It is not uncommon to hear the phrase "the Xth percentile," as percentile is another way of referring to 100-quantiles, or a distribution with 100 partitions.

Quantiles for Continuous Variables

Despite using quantiles to define a discrete set of data, they can be used to define continuous random variables as well. Because the data is continuous, the quantile uses an integral to define the variable. With a discrete data set, the

pth percentile is the number n for which p% of the data is less than n. Here, the function below is used to obtain a percentile for a continuous distribution where the pth percentile is a number n such that:

∫-₶n f ( x ) dx = p/100

The function f ( x

) is a probability density function, allowing for calculation of any percentile within a continuous distribution.

Applications of Quantiles

Quantiles allow for an understanding of a probability distribution of a data set in which only the specifications of the positions are known. A model, such as a normal distribution may apply, and quantiles of the data help inform which distribution model fits best. Quantiles from a data set can be compared with quantiles of a probability distribution model, like a Weibull distribution. The inferences from this comparison can be plotted in a scatterplot known as a quantile-quantile plot, or q-q plot. The suggested model may be a good fit for the data if the resulting scatterplot is generally linear.

By Iqr.png: Ark0nderivative work: Gato ocioso (talk) - Iqr.png, CC BY-SA 3.0,