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I know the difference between the mean and the median.

  • The mean of a set of numbers is the sum of all the numbers divided by the cardinality.
  • The median of a set of numbers is the middle number, when the set is organized in ascending or descending order (and, when the set has an even cardinality, the mean of the middle two numbers).

It seems to me that they're often used interchangeably, both to give a sense of what's going on in same data.

Do they mean (pun intended) different things? When should one be used over the other?

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    $\begingroup$ Knowing the median tells you that if you were to pick one of the data points at random, half of the time it will be below that value and half of the time it will be above (but it doesn't give you an idea of how much above or how much below). Knowing the mean on the other hand tells you the average result, so if you do many trials and add the results, you have a good idea of what the total would be (the median would not tell you that). $\endgroup$ – JMoravitz May 31 '17 at 20:15
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    $\begingroup$ If a country has a much lower median income than mean, then it probably has high inequality. If a country has a lot of income growth, but all the people benefitting are at the top, then you can conveniently hide the problem by talking about average income. If you want to highlight it, then you'll talk about the median income. $\endgroup$ – user49640 May 31 '17 at 20:29
  • $\begingroup$ @user49640 this should be an answer $\endgroup$ – Sevenate Mar 5 at 16:55
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Almost all analytic calculations on sets of data are more natural in terms of the mean than the median. For example, the "$z$-test for significance of a discrepancy relative to the null hypothesis deals with the sample estimated mean and sample unbiased estimated standard deviation.

The median, and particularly the difference between the median and the mean, is useful to characterize how "skewed" the data is (although the skew, which depends on the third moment about the mean, is also useful for that).

The real use of the median comes when the data set may contain extreme outliers (perhaps due to errors in early processing of the sample numbers, or a serious bias in the sample gathering procedure). Then describing the distribution in terms of quartiles (with the median dividing the second from the third quartile) can be more informative than quoting $\mu$ and $\sigma$.

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The median is particularly handy to describe data with a significant skew or long tail. For example, if we looked at incomes, a small number of rock-stars, corporate executives and hedge-fund managers each taking home multi-million dollar salaries. These outliers carry more weight in the calculation of the mean than they do in the median calculation. Mean income is higher than median income. The median income would be closer to something we associate with middle-class.

Means are great when the distribution has been well studied and is well understood. (e.g. normally distributed) Then mean and standard deviation tell us just about everything we care to know.

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  • $\begingroup$ I don't know what you mean when you say that the mean income would be just above the poverty line. According to this Wikipedia article ( en.wikipedia.org/wiki/… ), in 2014 the mean and median household incomes in the United States were respectively \$72,641 and \$53,718. For a family of three, the poverty threshold is about \$20,000. Something like what you're talking about might happen for individual incomes, but I doubt we'd be near the poverty line, especially if we don't count children. $\endgroup$ – user49640 May 31 '17 at 20:35
  • $\begingroup$ @user49640 you are absolutely right, I have the skewness backward on that one. The mean is distorted by the small number of multi-millionaires. I will edit the example. $\endgroup$ – Doug M May 31 '17 at 20:39

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