Sampling Distribution of S2

Sampling Distribution of $S^2$ (Sample Variance)

The sampling distribution of the sample variance $S^2$ is an important concept when making inferences about the population variance $\sigma^2$ based on sample data. It describes how the sample variance behaves when repeated samples are taken from the same population.

1. Sample Variance $S^2$ and its Calculation

The sample variance $S^2$ is a measure of the spread or dispersion of the sample data. It is calculated as:

Where:

• $x_i$ are the individual sample data points,
• $\bar{x}$ is the sample mean,
• $n$ is the sample size.

2. The Sampling Distribution of $S^2$

When we repeatedly take samples of size $n$ from a population and compute the sample variance $S^2$ for each sample, the distribution of these sample variances will follow a specific pattern, provided the population is normally distributed.

Key Properties:

1. Shape of the Sampling Distribution:

   • The sampling distribution of the sample variance $S^2$ follows a Chi-square distribution when the population from which the sample is drawn is normally distributed.
   • This distribution depends on the sample size $n$ and the population variance $\sigma^2$.
   • Specifically, if $X_1, X_2, \dots, X_n$ are independent and identically distributed (i.i.d.) random variables drawn from a normal population, then the sample variance $S^2$ is related to a chi-square distribution.

2. Degrees of Freedom:

   • The sampling distribution of the sample variance follows a Chi-square distribution with $n - 1$ degrees of freedom.
   • This is because the sample mean $\bar{x}$ is used to calculate $S^2$, and there is one degree of freedom lost when estimating the population mean from the sample.

3. Mean of the Sampling Distribution:

   • The mean of the sampling distribution of $S^2$ is equal to the population variance $\sigma^2$:
   $$E[S^2] = \sigma^2$$

   This means that the sample variance $S^2$ is an unbiased estimator of the population variance $\sigma^2$.

4. Variance of the Sampling Distribution:

   • The variance of the sampling distribution of $S^2$ is given by:
   $$\text{Var}(S^2) = \frac{2\sigma^4}{n - 1}$$

   This shows that the variability of the sample variance decreases as the sample size $n$ increases.

5. Standard Deviation of the Sampling Distribution (Standard Error of $S^2$):

   • The standard deviation of the sample variance is the standard error of the sample variance, which is:
   $$SE_{S^2} = \sqrt{\frac{2\sigma^4}{n - 1}}$$

   This formula helps us understand the spread of the sample variance estimates.

3. Chi-Square Distribution and the Sample Variance

As mentioned earlier, when the underlying population is normally distributed, the sample variance $S^2$ follows a chi-square distribution with $n - 1$ degrees of freedom.

The relationship between the sample variance and the chi-square distribution is:

Where:

• $\chi^2_{n-1}$ denotes a chi-square distribution with $n - 1$ degrees of freedom,
• $\sigma^2$ is the population variance,
• $S^2$ is the sample variance,
• $n - 1$ is the degrees of freedom.

This formula states that the scaled sample variance $\frac{(n-1) S^2}{\sigma^2}$ follows a chi-square distribution. The scaling factor $(n-1)$ accounts for the degrees of freedom and allows us to make inferences about the population variance.

Example:

If we have a sample of size 10 (i.e., $n = 10$) from a normally distributed population with a population variance $\sigma^2 = 25$, and we calculate the sample variance $S^2$, we can say that:

This means that the scaled sample variance $\frac{9S^2}{25}$ follows a chi-square distribution with 9 degrees of freedom.

4. Practical Use of Sampling Distribution of $S^2$

The sampling distribution of $S^2$ has several important applications in statistics:

1. Confidence Intervals for Population Variance:

   • We can use the chi-square distribution to construct confidence intervals for the population variance $\sigma^2$ based on the sample variance $S^2$.
   • For a given confidence level (say, 95%), the confidence interval for $\sigma^2$ can be calculated using the formula:
   $$\left( \frac{(n-1)S^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n-1)S^2}{\chi^2_{1-\alpha/2, n-1}} \right)$$

   Where:

   • $S^2$ is the sample variance,
   • $n$ is the sample size,
   • $\chi^2_{\alpha/2, n-1}$ and $\chi^2_{1-\alpha/2, n-1}$ are the chi-square critical values corresponding to the desired confidence level.

2. Hypothesis Testing:

   • The sampling distribution of $S^2$ is used in hypothesis testing for the population variance $\sigma^2$. For example, if we want to test whether the population variance is equal to a specific value $\sigma_0^2$, we can use the following test statistic:
   $$\chi^2 = \frac{(n - 1)S^2}{\sigma_0^2}$$

   This test statistic follows a chi-square distribution with $n - 1$ degrees of freedom.

Summary of Key Points

• The sampling distribution of $S^2$ (sample variance) is the distribution of the sample variance calculated from repeated samples taken from the population.
• If the population is normally distributed, the sampling distribution of $S^2$ follows a chi-square distribution with $n - 1$ degrees of freedom.
• The mean of the sampling distribution of $S^2$ is equal to the population variance $\sigma^2$, making $S^2$ an unbiased estimator of $\sigma^2$.
• The variance of the sampling distribution of $S^2$ is $\frac{2\sigma^4}{n - 1}$.
• The chi-square distribution plays a central role in constructing confidence intervals for the population variance and in hypothesis testing for variance.

Understanding the sampling distribution of $S^2$ is crucial when making inferences about population variances and performing statistical tests related to variance.