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Just been playing around with standard deviation and it seems it is always less than the range. Is this true in general, and if so, how could we prove it?

I do not really understand the answer to the duplicate question, and was looking for an answer that utilised the standard deviation formula.

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  • $\begingroup$ As stated, this is not true: if the range is $0$ so is the standard deviation, and it is false that $0<0$. $\endgroup$ – kimchi lover Dec 3 at 5:56
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To calculate the standard deviation you work out the deviation of each value in the dataset from the mean, square the deviations, find the mean of those squared deviations and then find the square root of that mean.

For each value, the deviation has to be less than or equal to the range. In fact it's only equal to the range if the value is at one extreme and the mean is at the other.

So let's state that $d<r$

Then $\Sigma d^2 < \Sigma r^2$

And $\frac {\Sigma d^2}n<\frac {nr^2}n$

Then $sd = \sqrt {\left(\frac {\Sigma d^2}n\right)}<\sqrt {\left(r^2\right)}$

So $sd < r$

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