# What is the median of the school grades A A B B ?

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?

• 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
Commented Jun 27, 2015 at 10:49
• 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. Commented Jun 27, 2015 at 10:58

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".