l> Stats: have the right to the typical deviation be much more than fifty percent of the range? (June 22, 2007)

Dear Professor Mean, ns was make the efforts to work-related with some simple data sets to view how huge I could make the typical deviation relative to the range. I know the typical deviation can never be bigger than the range, yet I can"t seem to obtain it to be bigger than half the range.

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There space some accountants who used to work with Enron who can do part very an innovative things with numbers. Have actually you tried talking to them?

It helps to look in ~ the formula. The most typically used formula is

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but some people use

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instead. If you usage the 2nd formula, climate it is pretty noticeable that the standard deviation cannot exceed the range. The typical of the data needs to be within the variety of the data, for this reason no single term (before being squared) in the sum can exceed the range. This makes it basic to show that the standard deviation needs to be smaller than the range.

If the data is symmetric, you can say also more. The median is specifically in the middle of the range. Each term in the sum is much less than or same to half the range, as such the typical deviation is much less than or equal to fifty percent the range.

You have the right to probably display that the conventional deviation is much less than or same to fifty percent the selection for asymmetric data together well, however the math is beyond my grasp on a Friday afternoon.

You can achieve the maximum traditional deviation because that a very simple case. Select two data points: 0 and 1 and also compute the traditional deviation making use of the second formula. It will certainly be 0.5, i m sorry is exactly half the range.

Now once you usage the very first formula, friend will obtain a slightly larger value. In fact the ratio of the very first formula come the 2nd formula is simply

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which is maximized for n=2. Us won"t talk about the situation where n=1. So the finest case (two data point out 0 and also 1) returns a conventional deviation that 0.7071 which is more than 50% the the range. Ns don"t think you deserve to do any much better than this.

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Oh yes, there is one an ext case to consider. Compute the typical deviation of the two data points 1 and 1. This gives you a traditional deviation the 0 and also a selection of 0. For this reason it"s feasible to get a standard deviation same to the range, but only for this one unique case.