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I know that if I calculate the median, I am going to get a number that represents central tendency even if there are outliers that would cause the distribution to be skewed right or left. According to the Aerd Statistics website:

When you have a normally distributed sample you can legitimately use both the mean or the median as your measure of central tendency. In fact, in any symmetrical distribution the mean, median and mode are equal.

To me, it seems like median is clearly useful in situations where there is skewed distribution. But mean seems to be a useless measurement since in symmetrical distribution it will be almost identical to the median.

My question is: why is mean used at all when in the case of symmetrical distribution a median represents central tendency just as well?

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  • $\begingroup$ If $X$ and $Y$ are random variables, is the median of $X+Y$ the sum of the medians of $X$ and $Y$? $\endgroup$
    – Angina Seng
    Oct 13, 2017 at 4:57
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    $\begingroup$ Why do you think the mean is only useful if you have a symmetric distribution? $\endgroup$
    – bof
    Oct 13, 2017 at 6:04
  • $\begingroup$ Added reasoning for thinking mean is only useful in symmetric distribution. I'm not saying I'm correct. Just notifying everyone that I provided the quote that led me to ask the question I did. $\endgroup$
    – Darien Springer
    Oct 13, 2017 at 6:35

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If your grades for the 7 classes you took last year were C,C,C,C,A,A,A, which is a pretty symmetric distribution, the median is C. You are a C student. But the guy who got 3 C's and 4 A's is an A student. Yet you're both very similarly talented students, with very close GPA's.

The type and purpose of the data (should) guide our choice of mean, median or mode. When we want to weight things, like grades, mean is a better measure. When we want to count things, median is better.

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  • $\begingroup$ What do you mean by weighing vs. counting? $\endgroup$
    – Darien Springer
    Oct 13, 2017 at 5:29
  • $\begingroup$ I'd say a good rule of thumb is if summing the values make sense or not in the context. The sum of your grades does make sense (it's the grade you get at a test that contains all the tests) so the mean makes sense. By contrast, if you're looking at the height of people, the sum of heights doesn't make much sense (it's the height of people stacked on top of each other, which doesn't seem useful) so the median is probably more relevant here. $\endgroup$
    – Joel Cohen
    Oct 13, 2017 at 6:49
  • $\begingroup$ Grades are weighted (A= 4, etc.) Salaries aren't (usually); we like to know how many (count) people are making more than we are. And in salaries, the median and mean can be very different numbers. (Largely because of the symmetry properties you mention.) $\endgroup$
    – B. Goddard
    Oct 13, 2017 at 14:11

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