What is the actual use of the median and mode in statistics compared to the average (mean)?
In: Mathematics
The better question to ask is….why do you even use mean in the first place? If your answer is “well….I need an aggregate quantity and I pick the mean because I was told to do so….”, that’s where the confusion came about.
Mean has specific usage, it’s not something that you just throw out there because you want a way to measure central tendency. In fact, it might seems like no-brainer today, but mean used to be quite controversial, until Gauss showed that it is actually useful. The more intuitive, default, quantity to measure central tendency used to be the *median*.
The purposes of these numbers are simple: they are the minimizers of “surprise”. Let’s say you were to need to use a single number to represent the value of every elements in the sample sets, but then you are given a single element, and if the element deviate from the number, you get punished by an amount dependent on how they differed. Then what you want is a number that minimize the expected amount of punishment.
Which number you use depends on how much punishment scale up with the deviation. The most intuitive measure is for the punishment to equal the deviation: the minimizer is the median, which is what people used for centuries. At an extreme end, any deviations at all is bad, the minimizer of this is the mode. And if the square of the deviation is the punishment, the minimizer is the mean.
The reason why the mean see a lot of use in theoretical statistics and probability theory is because: (a) by framing everything into a game theory context, lots of questions can be converted into about the mean of *something* (not necessarily the original data but something computed from it), so you don’t really lose generality by just studying the mean; (b) it has mathematically nice properties, such as linearity and central limit theorem.
But that doesn’t mean you should blindly use the mean in every situation. Think about what is your punishment. How bad it would be for your number to deviate from each samples? And this depends on practical context. If you are reporting numbers generally with no specific purposes, it might be useful to just report a whole bunch of different central tendency so that people have a choice of different numbers to make use of.
Latest Answers