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.
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.
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The sample standard deviation is biased, but it is consistent, meaning it approaches the true population standard deviation as the sample size increases.
Updated On: Dec 19, 2025
- unbiased and consistent
- biased and consistent
- unbiased and inconsistent
- biased and inconsistent
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The Correct Option is B
Solution and Explanation
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
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
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