Estimator: A statistic used to approximate a population parameter. Sometimes called a point estimator.Sheldon M. Ross
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 σ2.
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:
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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