Question

ERA Champions In 2014, Clayton Kershaw of the Los Angeles Dodgers had the lowest earned-run average (ERA is the mean number of runs yielded per nine innings pitched) of any starting pitcher in the National League, with an ERA of 1.77. Also in 2014, Felix Hernandez of the Seattle Mariners had the lowest ERA of any starting pitcher in the American League with an ERA of 2.14. In the National League, the mean ERA in 2014 was 3.430 and the standard deviation was 0.721. In the American League, the mean ERA in 2014 was 3.598 and the standard deviation was 0.762. Which player had the better year relative to his peers? Why?

Answer 1

Kershaw had the lower Z-score, this means that he had the better year.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by

`Z = (X - \mu)/(\sigma)`

The ERA is the Earned Runs Average per 9 innings. This mean that the lower the ERA is, the better it is. So, between Kershaw and Hernandez, whoever has the lower Zscore had the better season.

Kershaw

ERA of 1.77, so
`X = 1.77`.

In the National League, the mean ERA in 2014 was 3.430 and the standard deviation was 0.721. This means that
`\mu = 3.43, \sigma = 0.721`. So:

`Z = (X - \mu)/(\sigma)`

`Z = (1.77 - 3.43)/(0.721)`

`Z = -2.30`

Hernandez

ERA of 2.14, so
`X = 2.14`.

In the American League, the mean ERA in 2014 was 3.598 and the standard deviation was 0.762. This means that
`\mu = 3.598, \sigma = 0.762`.

`Z = (X - \mu)/(\sigma)`

`Z = (2.14 - 3.598)/(0.762)`

`Z = -1.91`

Kershaw had the lower Z-score, this means that he had the better year.