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

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

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

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

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 10-90 = 3.86 – 1.86 = 2

Percentile Exercise Homework 13