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08:31, 14 May 2007
[[User:SeanTheSheep|SeanTheSheep]] 03:05, 14 May 2007 (EDT)
The point about all of this is that calculations made on data sets are statistical estimates of something or other. If they are not, then there is no point to them. In this particular case, the average is an estimate of the central location or central tendency of the data, i.e. the answer to the question whereabouts is the data located?
A definition of average, which tells someone how to calculate one of the measures for this without explaining what we are trying to do, or discussing what the problems are in trying to capture such a thing is wrong.
The main reason that we almost uniformly use the arithmetic mean as the average is quite difficult to grasp: There is an important result in "Expectation Algebra" which says if you estimate the overall 'expected' mean of a probability distribution by using the arithmetic mean of a data set derived from that distribution, then the expected value of the distribution of arithmetic means is the mean that you started with; i.e. that the arithmetic mean is an '''unbiased estimator''' of the distribution mean (it is also a '''consistent estimator''', and, as it turns out the most '''efficient estimator'''). Those are the reasons it is used, and that result underpins what is probably the most important theorem in the whole of statistics: The Central Limit Theorem.
However, I am not suggesting that we clutter up the definition with all of that. What I am saying though is that there are lots of different ways of trying to find the middle of a set of data, and that saying that average is (add them up and divide by the number of numbers) is inapproprate on three counts:
* firstly that it ignores what the ''idea'' of average is trying to do,
* secondly that it does not discuss whether "average" defined in this manner actually captures the required notion
* thirdly it does not discuss in what situations an arithmetic average is, or is not appropriate.
--[[User:SeanTheSheep|SeanTheSheep]] 04:31, 14 May 2007 (EDT)
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