Concept About Estimated Standard Deviation

Question

I am looking for the concept about how to estimate standard deviation.

Actually I'm not sure how can I get a concept the estimate standard deviation ?

If you know the concept, then would you let me know it here?

Also if you have any reference or examples, it will be help to me.

p.s. Is this different between "estimate standard deviation" and "estimated standard deviation" ? I think it seems to the same.

Updated

What does different between "Method of Least Squares" and "estimate standard deviation"?

Are those the same concept? I don't know estimate standard deviation.

Answer 1

Given data $ { \left\{ {x}_{i} \right\} }_{i = 1}^{N} $ the Empirical STD of the data is well defined:

$$ STD = \sqrt{ \frac{1}{N - 1} \sum_{i = 1}^{N} { \left( {x}_{i} - \bar{x} \right) }^{2} } $$

Where $ \bar{x} $ is the empirical mean of the data given by:

$$ \bar{x} = \frac{1}{N} \sum_{i = 1}^{N} {x}_{i} $$

Now, if there's a model on the data (Such as signal and noise with certain CDF) the empirical calculation should be altered accordingly.

For instance, given a signal which is linear with AWGN the STD of the noise can be estimated by removing the linear estimated signal first.

Update

There are 2 classic estimator of the Standard Deviation (Also for Variance):

• The Unbiased Estimator: $ \sqrt{ \frac{1}{N - 1} \sum_{i = 1}^{N} { \left( {x}_{i} - \bar{x} \right) }^{2} } $.
• The Biased (Maximum Likelihood) Estimator: $ \sqrt{ \frac{1}{N} \sum_{i = 1}^{N} { \left( {x}_{i} - \bar{x} \right) }^{2} } $.