Each senior in a college course wrote a thesis. The lengths

Each senior in a college course wrote a thesis. The lengths in pages of those seniors's theses are summarized in the graph.

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 ?
Could someone provide a simpler solution to this problem ? The actual solution provided is quite confusing.

The graph gives us the following information:
No of pages : No of people
0-9 : 1
10 - 19 : 4
20 - 29 : 6
30 - 39 : 7
40 - 49 : 2

There are a total of 20 people (1+4+6+7+2)
The median number of pages will be the mean of the 10th and the 11th terms.
The 10th and 11th seniors lie in the 20-29 pages interval. So the median will lie in the 20 - 29 interval.

The list of the number of pages could look something like this:
List 1: 5, 10, 10, 10, 10, 20, 20, 20, 20, 20, 20, 30, 30, 30, 30, 30, 30, 30, 40, 40
or
List 2: 5, 10, 12, 12, 13, 20, 20, 20, 20, 29, 29, 39, 39, 39, 39, 39, 39, 39, 40, 40
or
List 3: 5, 10, 12, 12, 13, 20, 20, 20, 20, 27, 29, 36, 36, 36, 36, 36, 36, 39, 40, 40
or many many other ways

The highlighted terms are the 10th and the 11th terms.

a) What is the least possible number of seniors whose theses were within six pages of the median length ?

We need to move people away from the 10th and the 11th terms such that they are not within 6 pages of median.
Median can be 20 or 29 or anything in between (e.g. (28+29)/2 = 28.5 or (20+29)/2 = 24.5 etc)
In list 1, median is 20. In list 2, median is 29 (there are only two people within a distance of 6 from 29).
Is it possible to move ALL people 6 pages away from median? No. We have 4 more people in the 20-29 range and even if they all are at 20, the 10th person will be at 27 and 11th will be at 29. So the median will be 28. (see list 3)

So the 10th and the 11th person will always be within 6 pages of the median.

B) What is the greatest possible number of seniors whose theses were within six pages of the median length ?
We try to bring the people from all groups as close as possible to the median.
Only people from 10-19, 20-29 and 30-39 can possibly be 6 pages away. Let's bring the median right in the middle so that 10-19 and 30-39 can come within 6 pages of the median.

List: 5, 19, 19, 19, 19, 20, 20, 20, 20, 20, 29, 30, 30, 30, 30, 30, 30, 30, 40, 40
The median here will be (20+29)/2 = 24.5
Therefore we can have 4+6+7 = 17 seniors lie within 6 pages of the median.

There are total of 20 students.
1st group: 1 student wrote between 0-9 pages;
2nd group: 4 students wrote between 10-19 pages;
3rd group: 6 students wrote between 20-29 pages;
4th group: 7 students wrote between 30-39 pages;
5th group: 2 students wrote between 40-49 pages.

As there are 20 students, then the median length would be the average of 10th and 11th students writings. Both, 10th and 11th students are in the third group, so both these student wrote between 20 and 29 pages (so the median will be between 20 and 29).

A. We want to minimize the # of writings which fall in the range median-6 and median+6 pages. We want to spread the # of writings as far as possible. Make, all but 10th and 11th students from 3rd group to write 20 pages (min possible for this group), all student from 4th group to write 39 pages (max possible for this group) and 10th and 11th students to write 29 pages each, then median=(29+29)/2=29 and only these 2 student fall in the range 29-6=23 and 29+6=35 pages. Note that less than 2 is not possible as even if we put as far apart as possible 10th and 11th students' writings, 20 pages for 10th and 29th pages for 11th, then the median=(20+29)/2=24.5 and still these 2 student (at least) fall in the range median-6=18.5 and median+6=30.5 pages.

Answer: 2.

B. We want to maximize the # of writings which fall in the range median-6 and median+6 pages. We want to group # of writings as close as possible to the median, which still will be between 20 and 29. Now, make ALL 4 students from the second group to write 19 pages (max possible for this group) and ALL 7 students from the fourth group to write 30 pages (min possible for this group). Also make 10th and 11th students form the third group to write 24 and 25 pages, so that the median will be (24+25)/2=24.5. In this cases all these writings will fall in the range 24.5-6=18.5 and 24.5+6=30.5, so total of 4+7+6=17 writings.

Answer: 17.

Bunuel, I do not understand the explanation. I got median between 10-11th students median range of 20-29 pages. I lost you from there....request an explanation if possible to help me understand

A. In order to to minimize the # of writings which fall in the range median-6 and median+6 pages we should move as many writing as far as possible from the median. As the median is between 20-29 pages long (the third group) then move 4 students writings from this group to its lowest limit (20) and move all 7 students writings from the 4th group to its highest limit (39). Next, if 2 other students from the third group wrote 29 pages each, then the median would be 29, and only those two students will fall in the range from 29-6=23 and 29+6=35. Total: 2.

B. In order to to maximize the # of writings which fall in the range median-6 and median+6 pages we should move as many writing as close as possible from the median. The same here: move all 4 students writings from the 2nd group to its highest limit (19), move all 7 students writings from the 4th group to its lowest (30). Next, make the median so that median-6<19 (in order 4 students from the 2nd group to fall in the range) and median+6>30 (in order 7 students from the 4th group to fall in the range), which basically means that the median should be (19+30)/2=24.5. So, 10th and 11th students form the third group should write 24 and 25 pages (obviouslyall other students from this group will also be in the range). Total: 4+6+7=17.

Hope it's clear.

Sorry Bunuel, I still cannot get there. Your explanation is still not clear to me. I am very weak in statistics, if you could please break it down a bit more, perhaps I will get there. Thanks in advance

Can you please specify what part didn't you get?

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Hope it helps.

Hi Bunuel, I was not clear on how you were moving the students away and closer to the median. But Karishma already explained - got that totally now. Thank you Karishma. I will also review the links Bunuel mentioned now, I need to brush up statistics a lot....