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Comment by BeniBoy | Hacker News Reader

Definitely true!

But as in the precise situation you described, this can leads to discrepancies in result. So if you're coming from Matlab of R (or even Pandas), this can help to know that in Numpy the default is not the usual statistical one!

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I would not say there is any such thing as "the usual statistical" number for degrees of freedom. There are just different formulas for different estimators that have different properties (like bias) that sometimes matter in practical results and sometimes don't matter.

For example, in Bayesian statistics, summary statistics coming from a posterior distribution are _always_ biased, exactly according to the degree of bias encoded into the prior distribution, You _want_ results that are biased, so long as you believe your prior model of the bias accurately reflects the information you have available at the time.

If you chose to use a summary statistics like MAP in that setting, you would not care one bit whether it was biased or consistent or whatever. You'd just care that the posterior distribution is useful for a practical purpose.

In this sense, I think frequentist stats education falls short a lot of time time. There is nothing special about NHST as a framework for measuring significance of an effect. It's just one way to do things that sometimes is useful and other times isn't.

Similarly there is no special reason to ever care about BLUE estimators, unbiased estimators, efficient estimators, consistent estimators, etc. etc., or differences between different hypothesis tests like Wald test or Welch's t-test or Mann Whitney non-parametric test, yadda yadda yadda.

They are just different things with different properties and different formulas. None of them are "the usual statistical one." And any time you reference a software package using any of them, you should not expect the software package to have the same assumption about what default choices to make that you might believe from a textbook or something. There's no reason to expect them to be connected really at all.

But your estimate isn't very meaningful if you only have enough samples such that the N v (N-1) normalization matters, the variances on the estimator will be too large. The 1/N normalization comes from the MLE BTW, I'm sure most know this already. As an aside, to see the power of a biased estimator look at the James-Stein Estimator [1].

[1] https://en.wikipedia.org/wiki/James%E2%80%93Stein_estimator

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