by Marco Taboga, PhD
In this lecture we introduce and discuss the notion of quantile of the probability distribution of a random variable.
At the end of the lecture, we report some quantiles of the normal distribution, which are often used in hypothesis testing.
Table of contents
Informal definition
Problems with the informal definition
Problem 1  No solution
Problem 2  More than one solution
How to solve the problems
Formal definition
Example
Quantile function
Special cases
Special quantiles
Quantiles of the normal distribution
Other definitions
Informal definition
We start with an informal definition.
The quantile of a random variable is a value, denoted by , such that:

with probability ;

with probability .
Thus, the quantile is a cutoff point that divides the support of in two parts:

the part to the left of , which has probability ;

the part to the right of , which has probability .
Problems with the informal definition
There are important cases in which the informal definition works perfectly well. However, there are also many cases in which it is flawed. Let us see why.
Problem 1  No solution
In the above definition, we require that
Remember that the distribution function of a random variable is defined as
Therefore, we are asking that
However, we know that the distribution function may be discontinuous. In other words, it may jump and it may not take all the values between and .
As a consequence, there may not exist a value that satisfies equation (1). The distribution function may jump from a value lower than to a value higher than without ever being equal to .
Problem 2  More than one solution
The lack of existence of a solution to equation (1) is not the only problem.
In fact, not only the distribution function may jump, but it may also be flat over some intervals.
In other words, there may be more that one value of that satisfies equation (1).
How to solve the problems
How do we solve the two problems with the informal definition?
We start from problem 2: equation (1) may have more than one solution.
We could solve the problem by always choosing the smallest solution:
But this would leave problem 1 unsolved: since equation (1) may have no solution, the setmay be empty.
To solve both problems, we minimize over the larger set
Since any distribution function converges to as goes to infinity, the latter set is never empty (provided that ).
Therefore, we define the quantile as
Formal definition
What we have said can be summarized in the following formal definition.
Definition Let be a random variable having distribution function . Let . The quantile of , denoted by is
We have imposed the condition because:

if , then

if , then the setmay be empty, as, for example, in the important case in which has a normal distribution.
Example
Let us make an example.
Let be a discrete random variable with supportand probability mass function
The distribution function of is
Suppose that we want to compute the quantile for .
There is no such that
However, the smallest such that is because for and for .
Thus, we have
Quantile function
When is regarded as a function of , that is, , it is called quantile function.
The quantile function is often denoted by
Special cases
When the distribution function is continuous and strictly increasing on , then the smallest that satisfiesis the unique that satisfies
Furthermore, the distribution function has an inverse function and we can write
In this case, the quantile function coincides with the inverse of the distribution function:
Example If a random variable has a standardized Cauchy distribution, then its distribution function iswhich is a continuous and strictly increasing function. The quantile of is
Special quantiles
Some quantiles have special names:

if , then the quantile is called median;

if (for ), then the quantiles are called quartiles ( is the first quartile, is the second quartile and is the third quartile);

if (for ), then the quantiles are called deciles ( is the first decile, is the second decile and so on);

if (for ), then the quantiles are called percentiles ( is the first percentile, is the second percentile and so on).
Quantiles of the normal distribution
Some quantiles of the standard normal distribution (i.e., the normal distribution having zero mean and unit variance) are often used as critical values in hypothesis testing.
The quantile function of a normal distribution is equal to the inverse of the distribution function since the latter is continuous and strictly increasing.
However, as we explained in the lecture on normal distribution values, the distribution function of a normal variable has no simple analytical expression.
Therefore, the quantiles of the normal distribution need to be looked up in a table or calculated with a computer algorithm.
We report in the table below some of the most commonly used quantiles.
Name of quantile  Probability p  Quantile Q(p) 

First millile  0.001  3.0902 
Fifth millile  0.005  2.5758 
First percentile  0.010  2.3263 
Twentyfifth millile  0.025  1.9600 
Fifth percentile  0.050  1.6449 
First decile  0.100  1.2816 
First quartile  0.250  0.6745 
Median  0.500  0 
Third quartile  0.750  0.6745 
Ninth decile  0.900  1.2816 
Ninetyfifth percentile  0.950  1.6449 
Ninehundredth and seventyfifth millile  0.975  1.9600 
Ninetyninth percentile  0.990  2.3263 
Ninehundredth and ninetyfifth millile  0.995  2.5758 
Ninehundredth and ninetyninth millile  0.999  3.0902 
Other definitions
The above definition of quantile of a distribution is the most common one in probability theory and mathematical statistics.
However, there are also other slightly different definitions. For a review, see https://mathworld.wolfram.com/Quantile.html.
How to cite
Please cite as:
Taboga, Marco (2021). "Quantile of a probability distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentalsofprobability/quantile.