Population variance differs from sample variance in which of the following manner:
A. μ ± 3σ is replaced by x̄ + 9σ²
B. μ is replaced by x̄
C. μ² is replaced by μ² / (μ - 1)α
D. N is replaced by n - 1
E. N is replaced by n - 1 - α

Choose the correct answer from the options given below:

Correct option is A

The key difference between population variance and sample variance lies in the denominator and the mean used:

• Population variance uses the population mean (μ) and divides by the population size (N).

• Sample variance replaces μ with the sample mean (x̄), since the true mean is unknown.

• Sample variance divides by (n - 1), not n, to get an unbiased estimate of population variance.

So, B (μ replaced by x̄) and D (N replaced by n - 1) are the correct differences.

Information Booster:

Population variance measures variability of all data points in the population and uses the true mean and size. Sample variance estimates population variance from sample data, so it uses sample mean and corrects bias by dividing by (n - 1), called degrees of freedom. This correction is important especially for small samples to avoid underestimating the true variance.

Additional Knowledge:

• Statement A talks about empirical rules of distribution, not variance calculation.

• Statement C is not related to variance formulas.

• Statement E suggesting N replaced by n - 1 - α is incorrect and not used in variance calculation.