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Batting average in baseball is a statistic that represents the performance of a batter. It's calculated by dividing the number of hits by the number of at bats. For example, if a player has 20 hits out of 50 at bats, their batting average would be 0.400. This means that the player hits the ball successfully 40% of the time. A higher batting average means better performance at the plate, as it indicates the batter is successfully hitting the ball more often. For reference, a batting average of 0.300 and above is generally considered excellent in Major League Baseball (MLB).

Standard deviation is a measure of how spread out the numbers in a data set are. In baseball, it tells us how much the batting averages deviate from the mean (average). A low standard deviation means that the data points (batting averages) are close to the mean, while a high standard deviation means they are spread out over a wider range of values.
As a statistical tool, standard deviation is essential for comparing individual performances within a group. For example, knowing the standard deviation of batting averages, we can better understand how exceptional or ordinary someone’s performance is compared to others.

When comparing performances, especially in sports like baseball, we don't just look at straightforward statistics like raw numbers. Instead, using tools like the Z-score helps us understand how a player’s performance stacks up against others under similar conditions. The Z-score calculates how many standard deviations a particular value (like a batting average) is from the mean of the dataset.
In this example, Ted Williams' batting average of 0.406 had a Z-score of approximately 3.82, indicating he was almost 4 standard deviations above the mean. Meanwhile, Jose Altuve’s 0.341 batting average in 2014 had a Z-score of approximately 2.59, showing he was nearly 3 standard deviations above the mean.

• A higher Z-score means a performance is better relative to the average.
• It provides a way to compare across different years or leagues with different averages and deviations.

Since Williams had a higher Z-score, he had a better year relative to his peers compared to Altuve, even though their raw batting averages might suggest a simpler story.