Sample-quartile I don't know : Is there a sample such that

Nylah Hendrix 2022-07-14 Answered
I don't know : Is there a sample such that the mean does not lie between the lower and upper quartile? Is there a sample such that the median does not lie between the lower and upper quartile?
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Answers (1)

Tristin Case
Answered 2022-07-15 Author has 15 answers
"Is there a sample such that the mean does not lie between the lower and upper quartile?"
One such sample is { 1 , 2 , 3 , 4 , 5 , 120 }, which has quartiles Q 1 = 2 and Q 3 = 5, but a mean of 22.5. Note that the mean of a data set is heavily influenced by outliers while the quartiles are generally resistant to extreme values.
"Is there a sample such that the median does not lie between the lower and upper quartile?"
Consider the definitions of the median and quartiles. The first quartile of a data set separates the bottom 25 % of observations from the top 75 %. Similarly, the third quartiles separates the bottom 75 % of observations from the top 25 %. As noted in the comments, the median is simply the "second quartile" of a data set and separates the bottom 50 % from the top 50 %. If there was a data set that had a median less than the first quartile, then the observation at the 50th percentile would be less than an observation at the 25th percentiles, which makes no sense. Similar logic shows that the median cannot be greater than the third quartile.
This can be shown more rigorously by providing a formal definition for the median of a data set. We can define median x ~ of a set of n observations by
x ~ = 1 2 ( a n / 2 + a ( n / 2 ) + 1 )
where a i is the ith element of the order sequence of observations and : R Z is the ceiling function. The first quartiles is median of the sequence a 1 , . . . , a n / 2 and the third quartiles is the median of the sequence a ( n / 2 ) + 1 , . . . , a n . Hence, we have the ordering Q 1 x ~ Q 3 , meaning that the median can equal the first or third quartiles, but cannot be less than the first nor more than the third.

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