Refer to the chapter appendix in order to derive the mean of...

Refer to the chapter appendix in order to derive the mean of the sampling distribution of the sample variances for a sample of $n$ observations from a population of $N$ members when the population variance is $\sigma^2$. By appropriately modifying the argument regarding variances in the chapter appendix, show that $$ E\left[s^2\right]=N \sigma^2 /(N-1) $$ Note the intuitive plausibility of this result when $n=N$

Refer to the chapter appendix in order to derive the mean of the sampling distribution of the sample variances for a sample of $n$ observations from a population of $N$ members when the population variance is $\sigma^2$. By appropriately modifying the argument regarding variances in the chapter appendix, show that
$$
E\left[s^2\right]=N \sigma^2 /(N-1)
$$
Note the intuitive plausibility of this result when $n=N$

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Key Concepts

Sampling Distribution

The sampling distribution is the probability distribution of a statistic, such as the sample variance, computed from repeated samples drawn from a population. It forms the basis for studying the properties of estimators like the sample variance by examining their behavior over many samples.

Sample Variance

The sample variance measures the variability within a set of sample observations, and its estimation forms a critical part of statistical inference. Its calculation often involves adjustments to correct bias, especially when estimating the population variance from sample data.

Bias and Unbiased Estimation

Bias refers to the systematic error of an estimator, where an estimator is said to be unbiased if its expected value equals the true parameter. The derivation provided shows that the naive sample variance estimator is biased, and a correction by a factor of N/(N-1) is necessary to produce an unbiased estimator of the population variance.

Degrees of Freedom

Degrees of freedom represent the number of independent pieces of information available for estimating statistical parameters. When estimating the variance, using N-1 degrees of freedom instead of N corrects for the loss of a degree of freedom incurred by estimating the mean from the same data, leading to the unbiased formula provided.

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