Sheldon M. Ross

Estimator:A statistic used to approximate a population parameter. Sometimes called a point estimator.Estimate:The observed value of the estimator.Unbiased estimator:An estimator whose expected value is equal to the parameter that it is trying to estimate.

This post is based on two YouTube videos made by the wonderful YouTuber ** jbstatistics** : https://www.youtube.com/watch?v=7mYDHbrLEQo and https://www.youtube.com/watch?v=D1hgiAla3KI&list=WL&index=11&t=0s. The most pedagogical videos I found on this subject.

Sometimes, students wonder why we have to divide by *n-1* in the formula of the sample variance. In this pedagogical post, I show why dividing by *n-1* provides an unbiased estimator of the population variance which is unknown when I study a peculiar sample. I start with *n* independent observations with mean *µ* and variance *σ*^{2}.

I recall that two important properties for the expected value:

The variance is defined as follows:

Thus, I rearrange the variance formula to obtain the following expression:

For the proof I also need the expectation of the square of the sample mean:

Before moving further, I can find the expression for the expected value of the mean and the variance of the mean:

The expected value operator is linear:

I move to the variance of the mean:

Since the variance is a quadratic operator, I have:

Thus, I obtain:

I need to show that:

I focus on the expectation of the numerator, in the sum I omit the superscript and the subscript for clarity of exposition:

Because,

I continue by rearranging terms in the middle sum:

Remember that the mean is the sum of the observations divided by the number of the observations:

I continue and since the expectation of the sum is equal to the sum of the expectation, I have:

I use the results obtained earlier:

I wanted to show this:

I use the previous result to show that dividing by *n-1* provides an unbiased estimator:

The expected value of the sample variance is equal to the population variance that is the definition of an unbiased estimator.

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