Estimators – Advanced

Property 3: The sample variance is an unbiased estimator of the population variance

Proof: If we repeatedly take a sample {x1,…,xn} of size n from a population with mean μ, then the variance s2 of the sample is a random variable defined by

Sample variance

Since
image3136

it follows that

image3137 image3138

Since
image9002

and
image9003

by Property 1 of Expectation we now have

image3141 image3142

By Property 2 of Estimators

image3143

Putting all the pieces together we get

image3144

2 Responses to Estimators – Advanced

  1. Gilles says:

    Dear Charles,

    sorry, I’ve got another question…

    I do not understand how do we formally move from the definition of the variance sigma^2 = E[(x-µ)^2] to the one you’re using i nthe demo sigma^2 = sum(E[(x_i – µ)^2]).

    Thanks in advance,

    Gilles

    • Charles says:

      Dear Gilles,
      What was written on the referenced webpage was a typing mistake: sigma^2 = E[(x-µ)^2] without the sum. I have now corrected this mistake on the webpage. Thanks for catching the error.
      Charles

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