# Thread: Value that increases the Standard Deviation

1. ## Value that increases the Standard Deviation

Hello,

I am puzzled by the following statement

" In order to increase the standard deviation of a set of numbers, you must add a value that is more than one standard deviation away from the mean"

What is the proof of that? I know of course how we define the standard deviation but that part I seem to miss somehow. Any comments? Thanks!

2. ## Re: Value that increases the Standard Deviation

For a given set of numbers , the sample variance is given by

With a new additional number , the sample variance becomes

Then the difference is

and you see this is a quadratic expression in

To shorten our notation, let and . Then we can solve the quadratic inequality

So the width is not exactly 1 standard deviation as

Anyway the answer change if you, e.g. divide the sample variance by rather than

3. ## Re: Value that increases the Standard Deviation

Thanks BGM, that is indeed rigorous and to the point. But may I ask how you derive the last results? After completing the square and substituting for what I get is:

which is not exactly the same even after adding the fractions.

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