# Quantile of a probability distribution

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.

## Informal definition

The -quantile of a random variable is a value, denoted by , such that:

Thus, the quantile is a cut-off 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.

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
Twenty-fifth 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
Ninety-fifth percentile 0.950 1.6449
Nine-hundredth and seventy-fifth millile 0.975 1.9600
Ninety-ninth percentile 0.990 2.3263
Nine-hundredth and ninety-fifth millile 0.995 2.5758
Nine-hundredth and ninety-ninth 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.