Can Sample Variance Be Smaller Than Population Variance

Can Sample Variance Be Smaller Than Population Variance? A Deep Dive into Statistical Inference

Understanding the relationship between sample variance and population variance is crucial in statistics. While intuition might suggest that the sample variance is always smaller, the reality is more nuanced. This article explores the conditions under which sample variance can be smaller, larger, or equal to population variance, delving into the concepts of sampling error and unbiased estimators. We'll also examine how the degree of freedom impacts this relationship.

Understanding the Basics: Population vs. Sample Variance

Before diving into the core question, let's define our terms. Population variance represents the average squared deviation of each data point from the population mean. It's a measure of the overall spread or dispersion of the entire population. Sample variance, on the other hand, measures the spread within a randomly selected subset (sample) of that population. Calculating sample variance involves using the sample mean, which is an estimate of the true population mean.

The Role of the Sample Mean

The key to understanding the potential for sample variance to be smaller than population variance lies in the sample mean. The sample mean is an estimate of the population mean; it’s unlikely to be exactly the same. When the sample mean happens to be very close to the population mean, the calculated sample variance might be smaller than the true population variance. Conversely, if the sample mean significantly deviates from the population mean, this will increase the calculated sample variance.

Unbiased Estimators and Bessel's Correction

The standard formula for sample variance uses a denominator of n (sample size). However, this formula produces a biased estimator of the population variance; it tends to underestimate the true population variance. To address this bias, Bessel's correction adjusts the denominator to n-1, where n is the sample size. This adjusted formula provides an unbiased estimate of the population variance.

When Sample Variance Can Be Smaller

It's perfectly possible, especially with smaller sample sizes, for the sample variance calculated using n as the denominator to be smaller than the population variance. This occurs when the sample mean happens to be exceptionally close to the population mean, leading to smaller squared deviations and, consequently, a smaller variance. However, it’s important to note that this is a result of the biased estimator. Using Bessel’s correction usually makes it less likely for the sample variance to be smaller.

When Sample Variance Can Be Larger

More frequently, the sample variance (even with Bessel's correction) will be larger than the population variance. This is because sampling is inherently prone to error. The sample we select might inadvertently contain data points that are more spread out than the population as a whole, leading to an overestimation of the variance. This is explained by the concept of sampling error.

Degree of Freedom and its Influence

The concept of degrees of freedom is intricately linked to sample variance. When we use n-1 in the denominator (Bessel's correction), we are essentially accounting for the loss of one degree of freedom because we've used the sample mean to calculate the variance. This adjustment is particularly important for smaller sample sizes, as it significantly affects the accuracy of the variance estimate.

In Conclusion

While less frequent than the situation where sample variance exceeds population variance, it is possible for the sample variance to be smaller than the population variance. This is more likely with the biased estimator (denominator of n), particularly when the sample mean is exceptionally close to the population mean. However, with Bessel's correction and consideration of sampling error, the likelihood of a smaller sample variance decreases. The relationship is complex and influenced by factors such as sample size, the specific sample chosen, and the use of an unbiased estimator. Therefore, it is important to remember that sample variance provides only an estimate of population variance and should be interpreted with an awareness of the inherent uncertainties of sampling.