Sample-quartileI 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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Sample-quartileI don&#039;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?

Tristin Case · Accepted Answer

&quot;Is there a sample such that the mean does not lie between the lower and upper quartile?&quot;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.&quot;Is there a sample such that the median does not lie between the lower and upper quartile?&quot;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 &quot;second quartile&quot; 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.

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

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