The sample variance
a. could be smaller, equal to, or larger than the true value of the population variance.
b. is always larger than the true value of the population variance.
c. is always smaller than the true value of the population variance.
d. can never be zero.
The Correct Answer and Explanation is :
The correct answer is:
a. could be smaller, equal to, or larger than the true value of the population variance.
Explanation:
Variance is a measure of how spread out the values in a data set are. In statistics, we differentiate between the population variance and the sample variance. The population variance (denoted as ( \sigma^2 )) refers to the variance of an entire population, while the sample variance (denoted as ( s^2 )) is an estimate of the population variance based on a subset (sample) of that population.
- Sample variance could be smaller, equal to, or larger than population variance:
The sample variance ( s^2 ) is an estimate of the population variance ( \sigma^2 ). Since the sample is just a subset of the population, it might not capture the exact spread of the population data. Due to random sampling variations, sometimes the sample variance will be smaller than the population variance, sometimes it will be larger, and in rare cases, it could equal the population variance. - Unbiased estimator:
The sample variance is often computed using the formula that divides by ( n – 1 ) (where ( n ) is the sample size) instead of ( n ) to correct for bias. This is because using ( n – 1 ) compensates for the tendency of a sample to underestimate the true population variance. However, even though the formula is corrected, the sample variance still fluctuates due to sampling variability. - Sample variance can be zero:
The sample variance can be zero if all values in the sample are the same, meaning there is no variability among them. However, this is very rare in large samples unless the data are perfectly uniform.
Thus, the sample variance could be smaller than, equal to, or larger than the population variance, depending on the specific sample taken.