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Someone recently asked me the business use of various measures of central tendencies. Although the usage of mean and mode is intuitive and easily seen, I could not think of one intuitive use of median. What is median used for?

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Suppose you are forecasting future sales of whatever product you are producing. Suppose further that your bonus is tied to your forecasting performance, specifically on getting a low Mean Absolute Error (MAE, ) between the actual sales and your forecast.

It turns out that you can maximize your bonus by understanding the full predictive density and using its median as a point forecast.

Of course, if you assume your future sales to be (conditionally) normally or otherwise symmetrically distributed, then this median is the same as the mean or the mode. But if you believe future sales might be asymmetrically distributed, e.g., gamma or lognormally, then the difference between a median and a mean forecast might have an impact on your bonus paycheck. I'd say it does not get much more practical than that.

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Median is a measure of central tendency and defined as the middle point on a sorted array. Usually, there are outliers in your data, e.g. very high/low valued samples. In case you do not make any weighting on these samples during averaging, they will seriously affect the mean value. In contrast, since it operates on sorted data, median is robust against outliers. You can verify this as median values of $[1,2,3,4,5]$ and $[1,2,3,4,10]$ are identical.

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  • $\begingroup$ If your data are 0s and 1s with about equal numbers of each then the median could easily flip back and forth between 0, 0.5 and 1. In practice that is much less of a worry than outliers in one or both tails, but the median isn't robust in all circumstances. $\endgroup$ – Nick Cox Mar 2 at 9:20

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