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Sample variance is a measure used in statistics to quantify the amount of variation or dispersion in a sample dataset. It shows how much the data points in a sample differ from the sample mean. When you calculate sample variance, you're essentially looking at the average squared deviation of each observation from the mean of the sample.

A step-by-step approach to compute sample variance involves the following steps:

• First, sum all the observations in your sample to get \( \Sigma x_i \).
• Next, square each observation and compute the sum of these squares, denoted as \( \Sigma x_i^2 \).
• The number of observations in the sample is represented as \( n \).
• Now, substitute these values into the variance formula: \[ s^2 = \frac{\Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n}}{n - 1} \]

The formula shows how the sum of squares and sum of observations are crucial parts of this calculation. Remember that variance is expressed in squared units of the original data, providing a useful foundation for understanding total variation within a sample.

Sample standard deviation is closely related to variance, but it provides a more intuitive measure for most people to understand. While variance gives us the average squared deviation, the standard deviation provides this in the original units of the data, making it easier to interpret.

To find the sample standard deviation:

• First, calculate the sample variance using the method described above.
• Then, take the square root of the sample variance: \[ s = \sqrt{s^2} \]

Since the variance was 0.5257 in the exercise, the standard deviation becomes 0.7249 after taking the square root. This step gives a clearer indication of the amount of variation or dispersion present in the dataset relative to the mean.

The standard deviation is a useful statistic because it provides insight into the "average" distance data points are from the mean, allowing researchers to make generalized statements about how the values spread.

The sum of squares is a significant concept in statistics that plays a key role in variance and standard deviation calculations. It refers to the sum of the squared differences between each data point and the mean, and it is abbreviated as \( \Sigma x_i^2 \).

Calculating the sum of squares involves:

• Squaring each observation in the dataset to eliminate negative differences and highlight larger deviations.
• Adding these squared values together to get the total sum of squares.

The sum of squares is essential as it starts the process of variance calculation. In the exercise, we saw that \( \Sigma x_i^2 = 211.7421 \), demonstrating its pivotal role in determining how much variability is present in the data. Understanding the sum of squares helps to identify the spread and skewness, especially when coupled with other statistical measures like variance.