Question

which of the following is a biased estimator?
a. variance
b. proportion
c. standard deviation
d. mean

Explanation:

A biased estimator is one where the expected value of the estimator is not equal to the parameter it is estimating. The sample variance \(s^{2}=\frac{1}{n - 1}\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\) is an unbiased estimator of the population variance \(\sigma^{2}\). The sample mean \(\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}\) is an unbiased estimator of the population mean \(\mu\). The sample proportion \(\hat{p}=\frac{X}{n}\) (where \(X\) is the number of successes in \(n\) trials) is an unbiased estimator of the population proportion \(p\). However, the sample standard deviation \(s=\sqrt{\frac{1}{n - 1}\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\) is a biased estimator of the population standard deviation \(\sigma\). The bias in the standard - deviation estimator arises from the non - linear transformation (taking the square root) of the variance estimator.

Answer:

C. standard deviation