'n-1' versus 'n' in sampling variance

[Bignose]

I do want to know how you were using integrals to represent variance. I'm not familiar with integral notation yet.

As I wrote above, the LHS is from the definition of variance and the RHS is to represent the average of the sample variances.

 

In short, I go all the way back to the very first statement I wrote in reply to your calculations:

 

The average of sample variances, in general, will not be equal to the population variance. You showed this yourself with your calculations. To correctly combine sample variances, you need to follow the methodology provided in the link. In effect, you have to combine all the samples together into one, so the formula weights each sample variance by how may instances were in that sample + by how far the sample mean was from the combined sample mean.

 

The link shows how to do it for 2 samples, there exists generalizations for any number of samples.

[MonDie]

BigNose, I think you're overestimating me. I had a natural talent for math, but I've mostly studied science.

I'm going to take a month-long break from this forum, but I'll still see any follow-up posts.

 

I can understand the reasoning behind most of that equation, although I don't see why a sample's variance is being weighted relative to its mean, nor why they never "unsquare" for the squaring occurring inside te brackets, nor why their calculated combined variance was the same as the variance of the male sample. And I think it thus follows that I have no idea how it relates to Bessel's correction.

Nevertheless, I understood studiot's calculations, and I saw for myself that Bessel's correction works at least for multiple variances from multiple samples.

[studiot]

Mondie, you said you didn't want the algebra (you don't need calculus) to derive Bessel so I could only offer a worked example.

 

You can replace my numbers by symbols and work out the squares ( the algebra is little more than expanding (a+b)² for the general formulae.

[MonDie]

Mondie, you said you didn't want the algebra (you don't need calculus) to derive Bessel so I could only offer a worked example.

 

You can replace my numbers by symbols and work out the squares ( the algebra is little more than expanding (a+b)² for the general formulae.

 

Go ahead. I'll enjoy it when I get around to it.

[Sahil539]

This video explains the answer to your question in detail. It uses simulations to actually take samples to show whats wrong with using 'n'

 

https://www.youtube.com/watch?v=xslIhnquFoE

 

There is also part 2 and 3 to this video which go in deeper details, including the proof, in case you are interested.