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If the sample was $S_1=\{1,2,3,4\} \subset \mathbb R $ the median is 2.5 because 50% of the sample is $>=$ 2.5 and 50% of the sample is $<=$ 2.5. The median obviously does not need to be element of the sample, since 2.5 $\notin S_1$.

Assume the sample of ordinal school grades is $S_2 = \{A,A,B,B\}$ with $A$ better than $B$. What is the median?

  • $A$?
  • $B$?
  • $A$ or $B$?
  • $A$ and $B$?
  • The median is not defined?

For calculating the median of $S_2$ there is other letter $\notin S_2$ to be chosen from. Only $A$ and $B$ are potential candidates.

Can we conclude that the median is actually a not well-defined function?

HOSS_JFL
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    Isn''t the median of $\{1,2,3,4\}$ $2.14159265\dots$ because 50 % of the sample is $\leq 2.14159265\dots$ and 50 % of the sample is $\geq 2.14159265\dots$? – JiK Jun 27 '15 at 10:49
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    I know you are trolling here, but for an even number of metric (numerical) values the standard definition of the median applies with the mean of 2 and 3 here. For non-numerical values I do not see the definition.... and I heard the term "comedian" the first time not in the context of George Carlin et al. – HOSS_JFL Jun 27 '15 at 10:58

1 Answers1

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Both $A$ and $B$ would be valid medians, since at least half the data are $\leq A$ and $\geq B$; you could call one the "lower" median and one the "upper" median, or if uniquely defined medians are important you could try make the set that includes both "the median".

Glen_b
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