Chapter 12

12.1 There are 22 observations, so the median lies halfway between the 11th and 12th numbers. The middle two values are 35 and 41, so the median is

There are 11 observations to the left of the location of the median. The first quartile is the median of these 11 numbers and so is the sixth number. That is,

Q1 = 11

The third quartile is the median of the 11 observations to the right of the median’s location:

Q3 = 47

12.2

image

The median (38) and the third quartile (47) for Ruth are slightly larger than for Bonds and Aaron. The distribution for Ruth appears more skewed (left-skewed) than for Bonds and Aaron. If one examines Ruth’s career, one finds that he was a pitcher for his first six seasons, and during those seasons, he did not have many plate appearances. Hence, he has six seasons of very low home run counts, resulting in a left-skewed distribution.

12.3 To find the mean,

To find the standard deviation, use a table layout:

Observation Squared distance from
mean
13 (13 − 32.83)2 = (−19.83)2
= 393.2289
27 (27 − 32.83)2 = (−5.83)2
= 33.9889
⋮
10 (10 − 32.83)2 = (−22.83)2
= 521.2089
sum = 2751.3200

The variance is the sum divided by n − 1, which is 23 − 1, or 22.

630

The standard deviation is the square root of the variance.

The mean (32.83) is less than the median of 34. This is consistent with the fact that the distribution of Aaron’s home runs is slightly left-skewed.