Which of the following is a good point estimator for the population variance?

• A. The sample variance calculated as $\frac{\sum _{i=1}{(x_i-\overline{x})}^2}{n-1}$
• B. The sample standard deviation $s$
• C. The sample variance calculated as $\frac{\sum _{i=1}{(x_i-\overline{x})}^2}{n}$
• D. The sample mean $\overline{x}$

Answer: (A)

Verified step by step guidance

1. Understand that a point estimator is a statistic used to estimate a population parameter based on sample data.
2. Recall that the population variance is estimated by the sample variance, which accounts for degrees of freedom to avoid bias.
3. Identify the formula for the sample variance as: \(\displaystyle s^{2} = \frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}}{n - 1}\), where \(n\) is the sample size, \(x_{i}\) are the sample observations, and \(\overline{x}\) is the sample mean.
4. Note that dividing by \(n - 1\) (instead of \(n\)) corrects the bias in the estimation of the population variance, making this formula an unbiased estimator.
5. Recognize that the sample variance calculated with denominator \(n - 1\) is the good point estimator for the population variance, rather than the one with denominator \(n\), the sample standard deviation, or the sample mean.