Chapter 1.3, Problem 82E

(a)

To determine

To find: the median.

Answer to Problem 82E

The median is 306.

Explanation of Solution

Calculation:

The Median for the 7 defensive linemen is.

Median = M = 306

The median is the midpoint of the distribution. The number such that half the observations are smaller and half are larger. In the given data about half of the observations are above 306 and half are below 306.

Conclusion:

“Therefore, the median is 306.”

(b)

To determine

To find: the change in mean and median

Answer to Problem 82E

Themean decreases, the median is unaffected.

Explanation of Solution

Given:weighs 265 pounds instead of 285 pounds

Calculation:

If there had been 265 in place of 285 then mean and median would be:

Mean

  $\begin{array}{l}\overline{X}=\frac{\text{sum of observations}}{n}\\ \overline{X}=\frac{x_1+x_2+x_3+\mathrm{.....}+x_n}{n}\\ \overline{X}=\frac{{\displaystyle \sum X_i}}{N}\\ {\displaystyle \sum X}=\frac{306+305+315+303+318+309+265}{7}\\ =\frac{2121}{7}\\ \overline{X}=303\end{array}$

Median= M=306

From the above calculation we have seen that, with change in the extreme value mean has been affected but median is same as before. This signifies that mean is sensitive to extreme values in the data but median is not.

The mean will decease if one of the values is changed form 285 pounds to 265 pounds.

Since 285 is the smallest value in the data set, the median will not be affected if one of the values in changed from 285 pounds to 265 pounds.

Property: mean is influenced by outliers (because it decreases), while the median is unaffected by outliers (also called robust).

Conclusion: “Hence, the mean decreases, median is unaffected and the property is robustness.”