Prove that the sample variance S^2 defined by (7.2) is unbiased.

Name: Problem 15, Chapter 7: Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Uploaded: 2021-07-02T20:34:06-08:00
Duration: 1 min 36 s
Channel: Maxime Rossetti
Description: Prove that the sample variance S^2 defined by (7.2) is unbiased.

step 1 So SP, or the standard deviations pooled, P for pooled, is given by this blue equation here. Now

Prove that the sample variance S^2 defined by (7.2) is...

Key Concepts

Sample Variance

The sample variance is a statistical measure that quantifies the variability or dispersion of data points in a sample around their mean. It is commonly used to estimate the overall variance of the population from which the sample was drawn, using the discrepancies between individual sample observations and the sample mean.

Unbiased Estimator

An unbiased estimator is a statistical estimator whose expected value equals the true value of the parameter being estimated. This property ensures that, on average, the estimator neither overestimates nor underestimates the parameter, making it a desirable quality in statistical inference and estimation.

Bessel’s Correction

Bessel’s correction refers to the adjustment made in the calculation of the sample variance by dividing the sum of squared deviations by (n - 1) rather than n. This correction compensates for the bias introduced when the sample mean, an estimate of the true mean, is used in calculating the variability, thereby ensuring that the estimator is unbiased.

Degrees of Freedom

Degrees of freedom in the context of estimating variance indicate the number of independent values that can vary in the calculation after certain constraints, such as the computation of the sample mean. When the sample mean is calculated, one degree of freedom is lost, leaving n - 1 independent deviations, which is why the sample variance is computed with n - 1 in the denominator.

Expectation Operator

The expectation operator, or the concept of expected value, is a fundamental idea in probability that calculates the average outcome of a random variable over numerous repetitions of an experiment. In the context of proving the unbiasedness of the sample variance, the linearity of the expectation operator is applied to show that the expected value of the estimator equals the population variance.

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