Lower, Upper, and Inter quartile range of a continuous random variable

This article is a continuation from the previous article. This article explores the lower quartile, upper quartile and inter-quartile range of a continuous random variable;

Key Points

If X is a continuous random variable with cumulative distribution function (c.d.f) F(x) then the;

• lower quartile, Q₁ is given by; [IMAGE]
• upper quartile, Q₃, is given by; [IMAGE]
• inter-quartile range = upper quartile – lower quartile

Below we shall explore some examples;

Example

In this example we’re going to find the inter-quartile range;
Suppose a continuous random variable X has the cumulative distribution function;
[IMAGE]
We shall find the inter-quartile range for the function above; But first we must find the lower quartile and upper quartile; We know that;
[IMAGE]
So we’re looking for where F(Q₁)=0.25 We have to test the given ranges; when x=2, F(x)=0.2, but 0.2<0.25. So that must mean that the lower quartile lies in the range 21, which we’re looking for, so we have;
[IMAGE]
Let Q₁, so we have;
[IMAGE]
We can simplify by multiplying through by 10 to get;
[IMAGE]
…and solve to find Q₁
[IMAGE]
…so…
[IMAGE]
We want the value of Q₁ which is in the range of 23 to make it obvious that we’re finding the upper quartile range. So we evaluate the expression;
[IMAGE]
Remember we select the value that is in the estimated range. Here the range is 2