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Population Variance and Sample Variance


Given a population with size \(N\) and a sample from this population with size \(n\).

You can compute the exact value of population variance \(\sigma^2\) with:

$$ \sigma^2 = \frac1N \sum_{i=1}^N (x_i - \mu)^ 2 $$

where \(\mu\) is the mean of the population.

Second, you can compute the exact value of sample variance \(s^2\) with:

$$ s^2 = \frac1n \sum_{i=1}^n (x_i - \bar x)^ 2 $$

where \(\bar x\) is the mean of the sample.

Third, you can estimate the population variance based on the sample data when you don't know the mean of population:

$$ \sigma^2 \approx \frac1{n - 1} \sum_{i=1}^n (x_i - \bar x)^ 2 $$

where \(\bar x\) is the mean of the sample.

Finally, you can estimate the population variance based on the sample data when you know the mean of population:

$$ \sigma^2 \approx \frac1n \sum_{i=1}^n (x_i - \mu)^ 2 $$

where \(\mu\) is the mean of the population.

Ref:

  • Why divide the sample variance by N-1?

  • What is the difference between N and N-1 in calculating population variance?

  • Sample Variance

  • Standard Deviation and Variance

  • Variance



Published

Oct 19, 2018

Last Updated

Oct 19, 2018

Category

Tech

Tags

  • estimation 1
  • population 1
  • sample 1
  • variance 1

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