## 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).

## 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

*p*th 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

*p*th percentile is a number

*n*such that:

∫-₶n *f* ( *x* ) *dx* = *p*/100

*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.

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