ERA Champions In 2018, Jacob deGrom of the New York Mets 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 1.70. Also in 2018, Blake Snell of the Tampa Bay Rays had the lowest ERA of any starting pitcher in the American League with an ERA of 1.89. In the National League, the mean ERA in 2018 was 3.611 and the standard deviation was 0.772. In the American League, the mean ERA in 2018 was 3.744 and the standard deviation was 0.893. Which player had the better year relative to his peers? Why?

Verified step by step guidance

1. Step 1: Understand the problem context. We want to compare two players' ERAs relative to their respective leagues. Since the leagues have different average ERAs and variability, we need a standardized way to compare their performances.
2. Step 2: Recall that the z-score measures how many standard deviations a value is from the mean. The formula for the z-score is: \[ z = \frac{X - \mu}{\sigma} \] where \(X\) is the player's ERA, \(\mu\) is the league mean ERA, and \(\sigma\) is the league standard deviation.
3. Step 3: Calculate the z-score for Jacob deGrom using his ERA and the National League statistics: \[ z_{deGrom} = \frac{1.70 - 3.611}{0.772} \]
4. Step 4: Calculate the z-score for Blake Snell using his ERA and the American League statistics: \[ z_{Snell} = \frac{1.89 - 3.744}{0.893} \]
5. Step 5: Compare the two z-scores. The player with the lower z-score (more negative) performed better relative to his league peers because a lower ERA is better and a more negative z-score indicates a value further below the league average.

Key Concepts

Earned Run Average (ERA)

ERA measures the average number of earned runs a pitcher allows per nine innings pitched. It is a key statistic in baseball to evaluate a pitcher's effectiveness, with lower values indicating better performance. Understanding ERA helps compare pitchers' performances within and across leagues.

Standard Score (Z-Score)

A z-score indicates how many standard deviations a data point is from the mean. It standardizes values from different distributions, allowing comparison across groups with different means and variances. Calculating z-scores for ERA helps compare pitchers relative to their league peers.

Comparative Analysis Using Mean and Standard Deviation

Comparing individual performances requires considering the average (mean) and variability (standard deviation) of the group. By assessing how far a player's ERA deviates from the league mean in terms of standard deviations, we can determine who performed better relative to their peers.