Each senior in a college course wrote a thesis. The lengths, in pages, of those seniors' theses are summarized in the graph above.
a. What is the least possible number of seniors whose theses were within six pages of the median length?
b. What is the greatest possible number of seniors whose theses were within six pages of the median length ?
First determine the median thesis length. This seems a little tricky (since we have ranges for the thesis lengths), but we could still list the ranges in ascending order.
We get: (0-9),(10-19),(10-19),(10-19),(10-19),(20-29),(20-29),(20-29),(20-29),
(20-29),(20-29)...
Since there are 20 papers altogether (1+4+6+7+2 = 20), the median will be the mean (average) of the
10th and 11th terms.
So, the median can have several possible values.
a. What is the least possible number of seniors whose theses were within six pages of the median length?
Here's one way to minimize this value:
{2,11,11,11,11,20,20,20,20,
29,29,39,39,39,39,39,39,39,49,49}
Here, the median = (
29+29)/2 = 29.
So, there were only
2 theses within six pages of the median length
a. What is the greatest possible number of seniors whose theses were within six pages of the median length?
IMPORTANT: If we make the median length right in the middle of the 20-29 range, then the theses in the ranges on either side will both be within 6 pages.
Here's what I mean.
{2,19,19,19,19,20,20,20,20,
20,29,30,30,30,30,30,30,30,49,49}
Here, the median = (
20+29)/2 = 24.5
So, there were
17 theses within six pages of the median length
Cheers,
Brent
