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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 ? Answer: b. What is the greatest possible number of seniors whose theses were within six pages of the median length ? Answer: Attachment:
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abhi47 wrote: 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 ? The graph is attached as a jpg file . Could someone provide a simpler solution to this problem ? The actual solution provided is quite confusing. There are total of 20 students. 1st group: 1 student wrote between 09 pages; 2nd group: 4 students wrote between 1019 pages; 3rd group: 6 students wrote between 2029 pages; 4th group: 7 students wrote between 3039 pages; 5th group: 2 students wrote between 4049 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 median6 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 296=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 median6=18.5 and median+6=30.5 pages. Answer: 2. B. We want to maximize the # of writings which fall in the range median6 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 24.56=18.5 and 24.5+6=30.5, so total of 4+7+6=17 writings. Answer: 17.
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04 Feb 2012, 02:17
Bunuel, I do not understand the explanation. I got median between 1011th students median range of 2029 pages. I lost you from there....request an explanation if possible to help me understand
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sdas wrote: Bunuel, I do not understand the explanation. I got median between 1011th students median range of 2029 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 median6 and median+6 pages we should move as many writing as far as possible from the median. As the median is between 2029 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 296=23 and 29+6=35. Total: 2. B. In order to to maximize the # of writings which fall in the range median6 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 median6<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.
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07 Feb 2012, 00:28
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
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abhi47 wrote: 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 ? Attachment: WpS6j.png 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 09 : 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 2029 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, [highlight]20, 20[/highlight], 30, 30, 30, 30, 30, 30, 30, 40, 40 or List 2: 5, 10, 12, 12, 13, 20, 20, 20, 20, [highlight]29, 29[/highlight], 39, 39, 39, 39, 39, 39, 39, 40, 40 or List 3: 5, 10, 12, 12, 13, 20, 20, 20, 20, [highlight]27, 29[/highlight], 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 2029 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 1019, 2029 and 3039 can possibly be 6 pages away. Let's bring the median right in the middle so that 1019 and 3039 can come within 6 pages of the median. List: 5, 19, 19, 19, 19, 20, 20, 20, 20, [highlight]20, 29[/highlight], 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.
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07 Feb 2012, 08:31
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....
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10 May 2013, 22:53
A. 1.... the median itself. because the range in each column is 10, you could have everything but the median more than 6 pages away... if that's how it's meant to interpretted (6 pages is pretty arbitrary otherwise). For example, if the median is 29, and if the rest of column [2020] is only 20 pages each, and all of column [3039] is greater than 35 pages each. B. 17. If the median is 25 and all of column [1019] is actually 19, and all of column [3039] is actually 30...
I don't think this would be a GMAT question though.



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Excellent answers !



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21 Aug 2013, 11:45
I tried to calculate median using cumulative frequency and got 27.333 +  6 => 21.3 and 33.3. Using this range i got 0 and 13 answer. Attachment:
median.jpg [ 54.4 KiB  Viewed 9927 times ]
Is this method applicable to this question ? or I am using a wrong method please help to verify ? Thanks.
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21 Aug 2013, 22:15
PiyushK wrote: I tried to calculate median using cumulative frequency and got 27.333 +  6 => 21.3 and 33.3. Using this range i got 0 and 13 answer. Attachment: median.jpg Is this method applicable to this question ? or I am using a wrong method please help to verify ? Thanks. I am not sure what you have done here and why. How did you get a value for median? The median can take various different values. Also median is not a range or two points (21.3 and 33.3 that you obtained). Median of a data set is a single value. Are you somehow using Standard Deviation here?
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Hi Karishma, I found this formula to calculate median for data defined in intervals. Refer following link scroll down to median's formula. http://mbalectures.com/statistics/desc ... data.htmlI am just doing plus minus 6 pages to the median page count. In this range I can keep minimum 0 seniors and maximum all 13 seniors. Regards PiyushK
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26 Aug 2013, 22:54
PiyushK wrote: Hi Karishma, I found this formula to calculate median for data defined in intervals. Refer following link scroll down to median's formula. http://mbalectures.com/statistics/desc ... data.htmlI am just doing plus minus 6 pages to the median page count. In this range I can keep minimum 0 seniors and maximum all 13 seniors. Regards PiyushK The two concepts are different. In simple terms, the method you used is used to "best guess" the value of the median. Say, we only have the data of the graph and we need to guess the value of the median, then we can use that method. Actually, the value of the median might be much different. This question does not make you guess the value of the median. It says that the value can vary and asks for the case where you will get least/most members. It asks you how you should allot the pages so that you get the least/most value.
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10 Feb 2014, 04:51
Bunuel wrote: sdas wrote: Bunuel, I do not understand the explanation. I got median between 1011th students median range of 2029 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 median6 and median+6 pages we should move as many writing as far as possible from the median. As the median is between 2029 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 296=23 and 29+6=35. Total: 2. B. In order to to maximize the # of writings which fall in the range median6 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 median6<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. Hi Bunuel, I know the median length is the average length of the 10th and 11th theses, but in the group where all of the 6 theses have 2029 pages each, the 10th and the 11th theses are in the middle in terms of their lengths, therefore, given both 10th and 11th theses have 29 pages, the 8th and 9th theses in the 2029 group have 22 pages each , and all of the 7 theses in the 3039 group have 36 pages, I think the least number of the seniors writing theses satisfying the length within the range is 4, rather than 2, because the length of the 12th and 13th theses in the 2029 group cannot be shorter than the 10th and 11th theses in the same group. Am I right? Thank you.



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13 Feb 2014, 01:03
smallapple wrote: Bunuel wrote: sdas wrote: Bunuel, I do not understand the explanation. I got median between 1011th students median range of 2029 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 median6 and median+6 pages we should move as many writing as far as possible from the median. As the median is between 2029 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 296=23 and 29+6=35. Total: 2. B. In order to to maximize the # of writings which fall in the range median6 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 median6<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. Hi Bunuel, I know the median length is the average length of the 10th and 11th theses, but in the group where all of the 6 theses have 2029 pages each, the 10th and the 11th theses are in the middle in terms of their lengths, therefore, given both 10th and 11th theses have 29 pages, the 8th and 9th theses in the 2029 group have 22 pages each , and all of the 7 theses in the 3039 group have 36 pages, I think the least number of the seniors writing theses satisfying the length within the range is 4, rather than 2, because the length of the 12th and 13th theses in the 2029 group cannot be shorter than the 10th and 11th theses in the same group. Am I right? Thank you. No, you are not right. Consider example below: {0} {10, 10, 10, 10} {20, 20, 20, 20, 29, 29} {39, 39, 39, 39, 39, 39, 39} {40, 40} 1st group: 1 student wrote between 09 pages; 2nd group: 4 students wrote between 1019 pages; 3rd group: 6 students wrote between 2029 pages; 4th group: 7 students wrote between 3039 pages; 5th group: 2 students wrote between 4049 pages. Hope it helps.
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Re: Each senior in a college course wrote a thesis. The lengths [#permalink]
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21 Apr 2016, 04:03
# of people Pages 1 09 4 1019 6 2029 7 3039 2 4049
To consider is to stay within the boundaries of what is given as the groups. If I enter a new boundary then I will have to count the number of people inside the boundary. For instance, if my answer covers 31 pages to 19 pages, then my boundaries are 4, 6, and 7 = 13 people. These are in fact groups that I must add if my answer covers these number of pages
The median is the middle number. In this case we count the number of people total = 20. the middle median will be between the 10th and 11th number, both are found in the 3rd group.
1) least possible # of seniors whose theses 6 and +6 of the median length.
3rd group: 20,20, 29,29, 20,20 10th,11th
4th group: 39, 39, 39, 39, 39, 39, 39
we make the 3rd group median number to be the maximum number allowed for this group (29 out of the 2029 pages). In addition, make the 4th group all the maximum allowed of pages, or 39
the median is (29+29)/2 = 29. The least possible number of ppl whose theses are in 6 and +6 of the median length is 2, or the # of people in group 3:
296= 23 (stay within group 3 boundary) 29+6= 35 (stay within group 3 boundary of 2029)
One thing to consider is that because we did not change from group, meaning we are still in group 3 ( 2029) pages, we are working with the 6 people we know are in this group. Next, we also maximized the number of pages for four of the people ( 29, 29, 20, 20, 29, 29). The least possible number of seniors is 2 as a result.
stmt2 is to maximize the number of people within the Median +/6.
2nd group (use the maximum number of pages) 19, 19, 19, 19 3rd group _, _, 24, 25, _, _ 4th group (Use the minimum number of pages) 30, 30, 30, 30, 30 ,30, 30
average of the median (10th term + 11th term)/2 = (24 + 25)/2 = 24.5 pages
24.5 + 6 = 30.5 . this is breaking into the 4th group, basically we add 7 people at maximum. 24.5  6 = 18.5 . this is breaking to the 2nd group, add 4 people to your answer.
since you invaded group 4 and 2, you also have to count all the people in group 3. Final answer is 4+6+7 = 17



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20 Jan 2018, 17:28
Hello,
Can somebody explain how median of 10 and 11 fall into 6 students' pages? if we list students as following: 1 student  09 pages 2 students  4049 pages 7 students  3039 pages 4 students  1019 pages 6 students  2029 pages then, median of 1011 students won't correspond to 2029 pages.



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21 Jan 2018, 01:39
futurephilantropist wrote: Hello,
Can somebody explain how median of 10 and 11 fall into 6 students' pages? if we list students as following: 1 student  09 pages 2 students  4049 pages 7 students  3039 pages 4 students  1019 pages 6 students  2029 pages then, median of 1011 students won't correspond to 2029 pages. The median of a set with even number of elements is the average of two middle numbers when arranged in ascending or descending order. There are total of 20 students. 1st group: 1 student wrote between 09 pages, say 1 page each; 2nd group: 4 students wrote between 1019 pages, say 10 pages each; 3rd group: 6 students wrote between 2029 pages, say 20 pages each;4th group: 7 students wrote between 3039 pages, say 30 pages each; 5th group: 2 students wrote between 4049 pages, say 40 pages each. {1, 10, 10, 10, 10, 20, 20, 20, 20, 20, 20, 30, 30, 30, 30, 30, 30, 30, 40, 40}.As you can see, 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).
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Re: Each senior in a college course wrote a thesis. The lengths
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21 Jan 2018, 01:39



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