Which of the following statements about population variance is TRUE?

Question

BlackTom AI · Accepted Answer

Let’s break down what each option is claiming about population variance and test them against the fundamental definition of population variance.
Option a: 'Population variance measures data dispersion in the entire population.' This is correct. Population variance (often denoted as σ^2) quantifies how data points in the entire population are dispersed around the population mean. It is inherently a measure that describes the entire group, not just a sample.
Option b: 'Population variance is always zero.' This is incorrect. The variance would be zero only if all data points in the population were identical. In general, populations have some spread, so σ^2 is typically greater than zero.
Option c: 'Population variance is always less than sample variance.' This is not true as a universal rule. The relationship between population and sample variance depends on the data and the sample size. In fact, sample variance with denominator n-1 is an unbiased estimator of the population variance, which means its expected value equals the true σ^2, but a single sample variance can be larger or smaller than the true population variance for different samples. Therefore, one cannot claim it is always smaller.
Option d: 'Population variance uses n-1 in its calculation.' This is false. The population variance uses the denominator N (the population size). It is the sample variance that uses n-1 (the degrees of freedom) in its calculation to make it an unbiased estimator of the population variance when sampling is involved.
In summary, option a correctly describes population variance as a measure of dispersion for the entire population, while options b, c, and d contain common misconceptions about zero variance, comparative relations, and the use of n-1 in the population context.

Data Science Principles (202504-LLecture)

Which of the following statements about population variance is TRUE?

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