Consider the pooled variance as described in this wiki article https://en.wikipedia.org/wiki/Pooled_variance :

My observation is that the weighting terms

simply cancel out the Bessel's correction weighting factors. So then the result is simply

But then we no longer have count-based weightings on each of the variance terms! I must be missing something here - any insights/corrections?

Consider the pooled variance as described in this wiki article https://en.wikipedia.org/wiki/Pooled_variance :

My observation is that the weighting terms

simply cancel out the Bessel's correction weighting factors. So then the result is simply

But then we no longer have count-based weightings on each of the variance terms! I must be missing something here - any insights/corrections?

Consider the pooled variance as described in this wiki article https://en.wikipedia.org/wiki/Pooled_variance :

My observation is that the weighting terms

(n_i - 1) 

simply cancel out the Bessel's correction weighting factors. So then the result is simply

But then we no longer have count-based weightings on each of the variance terms! I must be missing something here - any insights/corrections?

Consider the pooled variance as described in this wiki article https://en.wikipedia.org/wiki/Pooled_variance :

My observation is that the weighting terms

simply cancel out the Bessel's correction weighting factors. So then the result is simply

But then we no longer have count-based weightings on each of the variance terms! I must be missing something here - any insights/corrections?