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 Math Central Quandaries & Queries
 Subject: Upper Quartiles Name: Jamie Who are you: Student I see you have a question about Q3 with even numbers but what about odd numbers? I have a problem with 19 numbers 36,45,49,53,55,56,59,61,62,65,67,70,75,81,82,86,89,94,99. Is there anyway the answer could be 81.5 because every time I do it I get 82 and my teacher tells me that is wrong. So in conclusion how do you do it?

Hi Jamie,

Quartiles can be confusing, especially since different authors and different textbooks have different definitions. I am going to explain using a process for finding the quartiles rather than using a definition in terms of percentages.

The second quartile Q2 (sometimes called the median) is easy to find if the data is in sorted order. If the number of observations is odd, as in your case, you take the middle observation, in your case 65. If the number of observations is even then Q2 is taken as half way between the middle two observations. So for example if your data set didn't include 99 and hence there were 18 observations then Q2 would be (62 + 65)/2 = 63.5.

To find Q3 you find the median of the upper half. Staying with the 18 observation data set above the upper half is 65,67,70,75,81,82,86,89,94 and the median of this data set is 81 so if the data set were 36,45,49,53,55,56,59,61,62,65,67,70,75,81,82,86,89,94 then Q3 = 81. With your 19 observation data set divided into two equal parts by the median

36,45,49,53,55,56,59,61,62,65,67,70,75,81,82,86,89,94,99

what is the upper half? Is it 67,70,75,81,82,86,89,94,99 or 65,67,70,75,81,82,86,89,94,99?
If it is the former then Q3, being the median of 67,70,75,81,82,86,89,94,99 is 82.
If the upper half is 65,67,70,75,81,82,86,89,94,99 then the median is (81 + 82)/2 = 81.5.

It seems that your textbook author and your teacher use the second process and get Q3 = 81.5

I hope this helps,
Penny

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