Question

Comparing batting averages Three landmarks of baseball achievement are Ty Cobb’s batting average of $0.420$ in $1911,$ Ted Williams’s $0.406$ in $1941,$ and George Brett’s $0.390$ in $1980$ These batting averages cannot be compared directly because the distribution of major league batting averages has changed over the years. The distributions are quite symmetric, except for outliers such as Cobb, Williams, and Brett. While the mean batting average has been held roughly constant by rule changes and the balance between hitting

and pitching, the standard deviation has dropped over time. Here are the facts:

Find the standardized scores for Cobb, Williams, and Brett. Who had the best performance for the decade he played?

Short answer

William performed best in the decade as the Williams’ $z-$score is the highest among all three.

Step-by-step solution

Given information

The table is:

Decade	Mean	Standard deviation
1910s	0.266	0.0371
1940s	0.267	0.0326
1970s	0.261	0.0317

Concept

The formula used:$z=(x-\mu )/\sigma$

Calculation

Standardized score:

$$\begin{gathered} ForCobb:ForWilliam:ForBrett\text{'}s: \\ z=\frac{0.42-0.266}{0.0371}z=\frac{0.406-0.267}{0.0326}z=\frac{0.390-0.261}{0.0317} \\ z=4.15z=4.26z=4.07 \end{gathered}$$

Because William's $z-$score is greater than Cobb's and Brett's, William's score fared best in the decade.