Question

Jerome's summer reading list has 8 books, and he is examining the number of pages in each book. After calculating the mean, median, standard deviation, and interquartile range, he realized that the longest book is actually 100 pages longer than he thought it was. Which of his measurements does he need to recalculate?

Answer 1

Basically the idea is to determine which measurements change when you change one element. Supose that the number of pages he thought the longest book had is represented by h

For instance, the average will change, since the new avarage will be

`\begin{gathered} \text{The average} \\ \bar{x}\text{ = }\frac{a_{}+b+c+d+e+f+g+(h_{}+100)}{8}, \end{gathered}`

This is also the case for the standard deviation, you can check the formula and see that it changes. However, the median and the interquartile range will not change.

For example, consider that the number of pages of the book at the beggining was

Then the mean is

`((100+110+120+130+140+150+160+170))/(8)=(1080)/(8)=135`

And the median is

`(130+140)/(2)=135`

But after he figured out that he have miscounted the number of pages of the last book, we have that the correct numbers of pages is

`100,110,120,130,140,150,160,170`

and the correct mean is

`((100+110+120+130+140+150+160+270))/(8)=(1180)/(8)=147.5`

so he had to recalculate the mean. On the other hand, the median is still the same, since the values of the middle, that is, 130 and 140 have not changed. Then, the median is

`(130+140)/(2)=135`