1. The Median of a Continuous Distribution
4.4
The Median of a Continuous Distribution
To calculate the median of a continuous distribution, we must use the cumulative distribution function F(x).
The probability that X has a lower value than the median is 0.5.
P(X ≤ x0.5) = F(x0.5) = 0.5 a b
F(x) x 1
0.5
x0.5

2. F(x) = 0 for x < 0 F(x) = for 0 ≤ x ≤ 4 F(x) = 1 for x > 4
The continuous random variable X is distributed with cumulative distribution function F where
F(x) = 0 for x < 0
F(x) = for 0 ≤ x ≤ 4
F(x) = 1 for x > 4
Find the median
F(x0.5) = 0.5

3. P(X ≤ x0.25) = F(x0.25) = 0.25 P(X ≤ x0.75) = F(x0.75) = 0.75
It is also possible to use the cumulative distribution function F(x) in order to calculate the inter-quartile range and the inter-percentile range of the distribution.
The inter-quartile range
The probability that X has a lower value than the lower quartile is 0.25.
The probability that X has a lower value than the upper quartile is 0.75.
P(X ≤ x0.25) = F(x0.25) = 0.25
P(X ≤ x0.75) = F(x0.75) = 0.75 a b
F(x) x 1
0.25
x0.25 a b
F(x) x 1
0.75
x0.75

4. The continuous random variable X is distributed with cumulative distribution function F where
F(x) = 0 for x < 0
F(x) = for 0 ≤ x ≤ 4
F(x) = 1 for x > 4
Calculate the inter-quartile range.
F(x0.75) - F(x0.25)
F(x0.25) =
F(x0.75) =
Inter-quartile range =
3.63 – 2.52 = 1.11

5. F(x) = 0 for x < 0 F(x) = for 0 ≤ x ≤ 4 F(x) = 1 for x > 4
The inter-percentile range
The continuous random variable X is distributed with cumulative distribution function F where
F(x) = 0 for x < 0
F(x) = for 0 ≤ x ≤ 4
F(x) = 1 for x > 4
Calculate the inter-percentile range 10-90
F(x0.9) - F(x0.1)
F(x0.1) =
F(x0.9) =
The inter-percentile range =
3.86 – 1.86 = 2

6. Percentile Exercise
Homework 13