Biased Estimator - Sample Variance

Hi Community,
Can someone please help me understand both intuitively and mathematically (if possible) why do we say that sample variance is a biased estimator of population variance and to overcome this biasness we need to divide sample variance by (n-1) instead of n,where n is the sample size.

Hi @VivekKapoor

The sample variance often underestimates the actual variance of the population. This happens because the sample mean is closer to the data points within the sample than the true population mean would be, leading to smaller squared differences. To correct this bias, we use Bessel’s correction: instead of dividing by the sample size n , we divide by n-1 .

This adjustment slightly increases the variance estimate, making it a better reflection of the true population variance.

Hope this helps!

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Hello, @VivekKapoor,

I do think this wiki page has given a very good mathematical explanation to the term n-1. It actually derives it step-by-step.

This thing is not quite intuitive, but if you would sit down and go through the wiki’s deviation, I would recommend you to try one thing:

In the first step, replace image with the population mean \mu. This is equivalent to asking “what will the sample variance be if I already had the population mean and did not need to compute the sample mean using the samples?”

Hopefully this will help you develop your own intuition about it.