Quantile Function

What is Quantile Function?

A Quantile Function is a phrase that describes a portion of a data set; it determines how many values in a distribution are above or below a given threshold. The quartile (quarter), quintile (fifth), and percentiles are special quantiles (hundredth).

The Q(a), is the inverse of the CDF; it gives us the probability that a random variable will be less than or equal to some value X = x.

Example of Quantile Function:

Two quantile measures are:

1. One is the value of the quantile function for 50%. This is termed the median of the distribution. On average, 50% of the variable’s outcomes will be below the median, and 50% of the variable’s outcomes will be above the median. The mean and median will be equal for a symmetric distribution (skew = 0). For distributions with positive (right) skew, the median will be less than the mean but greater for distributions with a negative (left) skew.
2. The second quantile measure of interest here is the interquartile range. The interquartile range is the upper and lower value of the outcomes of a random variable that include the middle 50% of its probability distribution. The lower value is Q(25%), and the upper value is Q(75%). The lower value is the value that we expect 25% of the outcomes to be less than, and the upper value is the value that we expect 75% of the values to be less than.

Why are quantiles important?

Quantiles provide information on the shape of a distribution, namely whether or not it is skewed. We can deduce that the distribution is skewed to the right if the higher quartile is further from the median than the lower quartile and vice versa.