The z-score is a measure of how many standard deviations a particular value is from the mean in a normally distributed data set. It is calculated using the formula:

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

Where:

• $x$ is the value for which we're calculating the z-score (in this case, 92.7 mph),
• $\mu$ is the mean (92.12 mph),
• $\sigma$ is the standard deviation (2.43 mph).

Substitute the values into the formula:

$z = \frac{92.7 - 92.12}{2.43}$

$z = \frac{0.58}{2.43} \approx 0.24$

So, the z-score for a pitch with a speed of 92.7 mph is approximately 0.24.

Would you like further details on how z-scores work or any related concepts? Here are five follow-up questions to consider:

1. What does a z-score of 0.24 signify in the context of the distribution of fastball speeds?
2. How would you interpret the z-score of a pitch significantly faster, say at 97 mph?
3. What percentage of pitches are expected to have speeds less than 92.7 mph based on the z-score?
4. How do different types of pitches (e.g., curveballs) compare in speed distribution to fastballs?
5. Why is it useful for statisticians to model pitch speeds using a normal distribution?

Tip: A z-score of 0 means the value is exactly at the mean of the distribution!