To determine which man had a more extreme height relative to the population, we can use the concept of z-scores. The z-score measures how many standard deviations a data point is from the mean. The formula to calculate a z-score is:

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

Where:

• $x$ is the individual data point (height in this case),
• $\mu$ is the mean of the population,
• $\sigma$ is the standard deviation of the population.

For the tallest man:

• $x = 246.246 \, \text{cm}$
• $\mu = 173.45 \, \text{cm}$
• $\sigma = 7.06 \, \text{cm}$

$z_{\text{tall}} = \frac{246.246 - 173.45}{7.06}$

For the shortest man:

• $x = 75.7 \, \text{cm}$
• $\mu = 173.45 \, \text{cm}$
• $\sigma = 7.06 \, \text{cm}$

$z_{\text{short}} = \frac{75.7 - 173.45}{7.06}$

Let's calculate these z-scores.The z-score for the tallest man is approximately 10.31, and for the shortest man, it is approximately -13.85.

Since the magnitude of the z-score indicates how extreme the height is relative to the population:

• The shortest man has a more extreme height because his z-score of -13.85 is further from zero compared to the tallest man's z-score of 10.31.

Would you like a detailed explanation of the z-score concept or any other clarifications?

Here are 5 questions to expand this topic:

1. How can we interpret positive and negative z-scores in general?
2. How can extreme z-scores affect data analysis?
3. How is the z-score useful in determining outliers?
4. What are some applications of z-scores in real-world data?
5. Can z-scores help in comparing different data distributions?

Tip: Z-scores are highly useful for comparing data points from different distributions by standardizing them!