91Ó°ÊÓ

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: $$ \begin{array}{clc} \hline \text { Decade } & \text { Mean } & \text { Standard deviation } \\ 1910 \mathrm{~s} & 0.266 & 0.0371 \\ 1940 \mathrm{~s} & 0.267 & 0.0326 \\ 1970 \mathrm{~s} & 0.261 & 0.0317 \\ \hline \end{array} $$ Find the standardized scores for Cobb, Williams, and Brett. Who was the best hitter?\(^5\)

Step by step solution

01

Understand the Z-Score Formula

The z-score is a measure of how many standard deviations an element is from the mean. The formula to compute the z-score is given by \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the player's batting average, \( \mu \) is the mean batting average of the decade, and \( \sigma \) is the standard deviation of that decade.

02

Calculate Cobb's Z-Score

For Ty Cobb in 1911, we use his batting average \( x = 0.420 \), the mean \( \mu = 0.266 \) from the 1910s, and the standard deviation \( \sigma = 0.0371 \). Substituting these into the formula gives us Cobb's z-score: \( z = \frac{0.420 - 0.266}{0.0371} \approx 4.15 \).

03

Calculate Williams's Z-Score

For Ted Williams in 1941, using his batting average \( x = 0.406 \), the mean \( \mu = 0.267 \) from the 1940s, and the standard deviation \( \sigma = 0.0326 \), we calculate his z-score: \( z = \frac{0.406 - 0.267}{0.0326} \approx 4.26 \).

04

Calculate Brett's Z-Score

For George Brett in 1980, using his batting average \( x = 0.390 \), the mean \( \mu = 0.261 \) from the 1970s, and the standard deviation \( \sigma = 0.0317 \), his z-score is calculated as: \( z = \frac{0.390 - 0.261}{0.0317} \approx 4.08 \).

05

Compare Z-Scores to Determine the Best Hitter

Now that we have the z-scores for all three players: Cobb (\( z \approx 4.15 \)), Williams (\( z \approx 4.26 \)), and Brett (\( z \approx 4.08 \)), we can compare them. A higher z-score indicates the player was more exceptional compared to their contemporaries. Ted Williams has the highest z-score, so he was the best hitter relative to the mean and standard deviation of his era.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Batting Average

Batting average is a widely used statistic in baseball that allows you to measure a player's hitting performance. In simple terms, it is calculated by dividing the number of hits by the number of at-bats. For instance, if a batter has 25 hits out of 100 at-bats, his batting average would be 0.250. The higher the average, the more successful a player is considered when hitting.

However, comparing batting averages from different eras directly is not always accurate. This is because the overall performance level can shift due to various changes in the game such as pitching styles, equipment, or rules. For this reason, within the broader context of statistical analysis, such averages must be interpreted carefully and often require additional statistical methods to make accurate comparisons.

Standard Deviation

Standard deviation is a measure used in statistics to quantify the amount of variation or dispersion in a set of values. In the context of batting averages, it captures how spread out the performances are across players within a specific timeframe.

When the standard deviation is larger, it implies greater variability in player performances during that era, which means players' performance levels were more widely spread out. Conversely, a smaller standard deviation suggests more consistency in players’ batting averages. This is why it’s essential to consider standard deviation when comparing players like Ty Cobb, Ted Williams, and George Brett. In baseball statistics, it provides a reference point to see how unusual or remarkable a player's average is relative to other players in their respective decades.

Statistical Comparison

For effective statistical comparison, especially across different eras, the z-score is often utilized. Z-score helps standardize different data points, making it easier to compare individual performances against their contemporaries irrespective of era-specific conditions.

To calculate the z-score, the formula used is: \( z = \frac{x - \mu}{\sigma} \), where \( x \) represents the individual score (in this case, the player’s batting average), \( \mu \) is the mean value of the set, and \( \sigma \) is the standard deviation. Applying this formula to Ty Cobb, Ted Williams, and George Brett allows comparing their batting averages by effectively removing era-specific differences in average and variance. Thus, each player's performance can be seen within the context of their contemporaries’ performances, helping better identify which player truly stood out as the best hitter.