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, thepth 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
) 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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