# Thread: Can we find minimum and maximum of SD?

1. ## Can we find minimum and maximum of SD?

Hello,
I have a histogram and calculate standard deviation.
Can we determine minimum and maximum of possible SD?
Thanks

2. ## Re: Can we find minimum and maximum of SD?

Hey life24.

You need to supply a distribution and set of constraints.

Remember that if sigma is greater [or equal] than zero then it can be anything finite which means there is no real maximum.

Is there any other information you want to supply?

3. ## Re: Can we find minimum and maximum of SD?

Originally Posted by chiro
Hey life24.

You need to supply a distribution and set of constraints.

Remember that if sigma is greater [or equal] than zero then it can be anything finite which means there is no real maximum.

Is there any other information you want to supply?
Thank you, distribution can be anything.

4. ## Re: Can we find minimum and maximum of SD?

You pretty much have to maximize the standard deviation function and that will depend on what the range of the distribution is along with the family of distributions that has a maximum standard deviation.

You will have to find a distribution where E[X^2] - E[X]^2 is maximized to answer this question.

5. ## Re: Can we find minimum and maximum of SD?

Originally Posted by chiro
You pretty much have to maximize the standard deviation function and that will depend on what the range of the distribution is along with the family of distributions that has a maximum standard deviation.

You will have to find a distribution where E[X^2] - E[X]^2 is maximized to answer this question.
Thank you, but we don't predict what is distribution. Can we determine maximize the SD without distribution?

6. ## Re: Can we find minimum and maximum of SD?

You will have to find a function f(x) [probability distribution] where you maximize that function of the variance.

So you have the following constraints:

Integral f(x)dx = 1 and

E[X^2] - E[X]^2 = Integral x^2*f(x)dx - [Integral x*f(x)dx]^2 is maximum.