Question

If \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2 \) is an unbiased and consistent estimator of the population variance, then one can conclude that \( S = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2} \) is an_________ estimator of the population standard deviation.

• A. unbiased and consistent
• B. biased and consistent
• C. unbiased and inconsistent
• D. biased and inconsistent

Hint:

The sample standard deviation is biased, but it is consistent, meaning it approaches the true population standard deviation as the sample size increases.

Answer 1

The Correct Option is B

Step 1: Understanding the relationship between the sample variance and standard deviation.
The sample variance \( S^2 \) is an unbiased and consistent estimator of the population variance. However, the sample standard deviation \( S \), being the square root of \( S^2 \), is generally biased.
Step 2: Evaluate consistency.
Although \( S \) is biased, it is still a consistent estimator of the population standard deviation because as the sample size increases, the bias diminishes, and the estimator approaches the true value.
Step 3: Conclusion.
Since \( S \) is biased but consistent, the correct answer is (B). Final Answer: (B) biased and consistent