Inspired by a question about standard dev, this one is a question that has been bugging me for years. I’ve been using the mean and stdev a lot in my work and during my college years, but I never really used median and mode outside when it was introduced. I’ve seen no use for it so far. So, ELI5? especially to those people who use this frequently.
In: Mathematics
Most of the time in math the word “average” is treated as interchangeable with “arithmetic mean.” But in more casual language, the word “average” is defined as the value/outcome that is most typical or normal.
For a lot of data sets, using the arithmetic mean as the average works out just fine, because the mean value is a good predictor of what you’d expect a typical element of the data set to be. But this isn’t always the case.
Consider of group of you, 9 of your friends, and Bill Gates. If you wanted to compute the average net worth of this sample and you tried to use the arithmetic mean as the average, you’d find that Bill Gates’ presence in the sample majorly skyrockets the average up into the billions. And, unless you have some *very* rich friends, the arithmetic mean is hardly representative of the net worth of a typical member of the sample,
On the other hand, if you calculated the average by using the **median**, you’d find that the average value is the middle value, such that five people have more money than this value and five people have less money. Since it’s likely that you and your friends have similar net worths, the median is a much better predictor of the typical net worth of this sample.
Now suppose you go to the store and you buy 100 boxes of donuts. Each of these boxes is advertised as having a dozen donuts, but 15 of them are defective – a hungry store worker ate a few from the box before resealing it and shelving it – and have fewer than a dozen.
Now let’s say that, just as curiosity, you want to know how many donuts are in an average box. You certainly could use the arithmetic mean or maybe the median as your “average”, but neither of these would really accurately describe a typical box of donuts. Here you’d want to use the **mode**, defined as the most frequently occurring element of the data set, since 85% of the boxes have 12 donuts.
TL;DR: The median is used as the “average” when most of the data is clustered together but there’s one or more large outliers; and mode is used as the “average” when a particular value is repeated a lot.
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