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The sample variance measures how much the individual data points in a sample differ from the sample mean. It gives an estimation of the spread or dispersion of the sample data. To calculate the sample variance, follow these steps:

• First, find the mean (\bar{x}) of the sample. This is the sum of all sample values divided by the number of samples.
• Next, subtract the mean from each data point to obtain the deviation of each point from the mean.
• Square each deviation.
• Sum up all the squared deviations.
• Finally, divide this sum by the number of data points minus one (n-1). This gives you the sample variance (s^2).

Using our example data (7, 8, 2, 10, 6, 5, 7, 8, 8, 9, 5, 9), the mean is 7. Subtract the mean from each data point, square the result, sum these squared deviations, and divide by 11 (since there are 12 data points):
\[ s^2 = \frac{1}{11} \sum (x_i - \bar{x})^2 \approx 4.91. \]

The sample standard deviation is a measure of the dispersion of the sample data points around the sample mean, representing a degree of variability within the sample. It is the square root of the sample variance. It is calculated as follows:

• First, find the sample variance (s^2).
• Then take the square root of the sample variance.

For our example, since the sample variance is approximately 4.91, the sample standard deviation will be:
\[ s = \sqrt{4.91} \approx 2.22. \]

The chi-squared distribution is a probability distribution that is used to estimate the variance of a population based on a sample. This distribution is particularly useful in hypothesis testing and constructing confidence intervals. Its shape depends on the degrees of freedom. When calculating a confidence interval for the standard deviation of a population, critical values from the chi-squared distribution are used to determine the bounds.
In our example, we want a 95% confidence interval for the standard deviation. We use the chi-squared distribution with 11 degrees of freedom to find critical values for chi-squared at 0.025 and 0.975.
These critical values help establish the lower and upper bounds of the confidence interval:
\( \text{Lower bound} = \frac{(n-1) s^2}{\chi^2_{0.975}} \text{ and } \text{Upper bound = } \frac{(n-1) s^2}{\chi^2_{0.025}}.\)

Degrees of freedom (df) refers to the number of independent values or quantities that can be assigned to a statistical distribution. In the context of calculating the sample variance and standard deviation, the degrees of freedom are equal to the sample size minus one (n-1).
Degrees of freedom are critical in determining the appropriate distribution (such as the chi-squared distribution) to use for statistical inferences.
In our example, the sample size is 12, so the degrees of freedom is:
\[ df = 12 - 1 = 11. \]

The sample mean (\bar{x}) represents the average value of the sample data. It is calculated by summing up all the sample values and then dividing by the total number of samples. The sample mean is a primary measure of central tendency in statistics.
Calculating the sample mean involves:

• Adding up all sample values.
• Dividing this sum by the number of samples (n).

Using our sample data (7, 8, 2, 10, 6, 5, 7, 8, 8, 9, 5, 9), the sum of the samples is 84 and the number of samples is 12. Therefore, the sample mean is:
\[ \bar{x} = \frac{84}{12} = 7. \]