To compare the performances of Player A and Player B, we can use z-scores, which help standardize their home run counts based on the mean and standard deviation of their respective seasons.

The z-score is calculated using the formula:

$z = \frac{{X - \mu}}{{\sigma}}$

Where:

• $X$ is the observed value (the number of home runs hit),
• $\mu$ is the mean of the population (mean home runs for that season),
• $\sigma$ is the standard deviation of the population (standard deviation of home runs for that season).

For Player A:

• $X_A = 64$ (Player A's home runs),
• $\mu_A = 18.3$ (mean home runs in Player A's season),
• $\sigma_A = 12.77$ (standard deviation in Player A's season).

So, the z-score for Player A is:

$z_A = \frac{{64 - 18.3}}{{12.77}} = \frac{{45.7}}{{12.77}} \approx 3.58$

For Player B:

• $X_B = 57$ (Player B's home runs),
• $\mu_B = 20.7$ (mean home runs in Player B's season),
• $\sigma_B = 13.43$ (standard deviation in Player B's season).

So, the z-score for Player B is:

$z_B = \frac{{57 - 20.7}}{{13.43}} = \frac{{36.3}}{{13.43}} \approx 2.70$

Conclusion:

• The z-score for Player A ($z_A \approx 3.58$) is greater than the z-score for Player B ($z_B \approx 2.70$).
• Therefore, Player A's home run feat was more impressive than Player B's based on their z-scores.

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Do you want more details or have any questions about this?

Here are 5 related questions:

1. What does a higher z-score indicate in terms of performance?
2. How would the comparison change if the standard deviations were larger?
3. What other statistical measures can be used to compare performances?
4. How does using z-scores help in comparing players from different seasons?
5. Can the z-score approach be used to compare other statistics like RBIs or batting averages?

Tip: Z-scores are especially useful when comparing performances across different seasons, as they account for varying averages and dispersions in the data.