Question

In Exercises 13–16, use z scores to compare the given values.

Tallest and Shortest Men The tallest living man at the time of this writing is Sultan Kosen, who has a height of 251 cm. The shortest living man is Chandra Bahadur Dangi, who has a height of 54.6 cm. Heights of men have a mean of 174.12 cm and a standard deviation of 7.10 cm. Which of these two men has the height that is more extreme?

Short answer

The height of the shortest man, equal to 54.6 cm, is more extreme than the height of the tallest man, equal to 251 cm.

Step-by-step solution

Given information

The mean height of men is given as 174.12 cm, and the standard deviation of the heights of men is equal to 7.10 cm.

Formula of z-score

Thez-score is the value that explains the location of a given data value from the mean value in terms of the standard deviation. Mathematically,

$Z=\frac{x-\overline{x}}{s}$

Values with a greater z-score (in absolute terms) are said to be less extreme compared to values with a smaller z-score.

Calculation

The z-score for the tallest man with a height equal to 251 cm is computed as follows:

$$\begin{gathered} z=\frac{x-\overline{x}}{s} \\ =\frac{251-174.12}{7.10} \\ =10.83 \end{gathered}$$

Therefore, thez-score for the tallest man with a height equal to 251 cm is equal to 10.83.

The z-score for the shortest man with a height equal to 54.6 cm is computed as follows:

$$\begin{gathered} z=\frac{x-\overline{x}}{s} \\ =\frac{54.6-174.12}{7.10} \\ =-16.83 \end{gathered}$$

Therefore, thez-score for the tallest man with a height equal to 54.6 cm is equal to -16.83.

Comparison

The height of the tallest man is 10.83 standard deviations above the mean, and the height of the shortest man is 16.83 standard deviations below the mean.

As 54.6 cm is farther from the mean value compared to 251 cm, theshortest man’s height, equal to 54.6 cm, ismore extreme than the tallest man’s height.