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The mean, also referred to as the average, is a measure of central tendency. It provides a quick summary of a dataset and helps understand the "middle" value. To calculate the mean:

• Add up all the data values in the sample.
• Divide the sum by the number of data points.

In the given exercise, the data values are 10, 20, 12, 17, and 16. Notice how each number contributes equally to the final calculation. When you sum these values, you get 75. Since there are five values in total, you divide the sum by 5, resulting in a mean (\( \bar{x} \)) of 15.
This calculation is fundamental in statistics for determining the average value of a dataset, which serves as a foundation for further statistical analysis like calculating the variance and standard deviation.

Standard deviation is a crucial statistical measure that quantifies the amount of variation or dispersion in a set of data values. In simpler terms, it tells you how spread out the numbers are in your dataset.
To calculate the standard deviation:

• Find the mean of the data set.
• Subtract the mean from each data value to find the deviation.
• Square each of these deviations.
• Find the average of these squared differences to calculate the variance.
• Take the square root of the variance to arrive at the standard deviation.

For our data values, after subtracting the mean (15) from each number and squaring the result, the squared differences are: 25, 25, 9, 4, and 1. These are then summed to give a total of 64.
The variance (\( \sigma^2 \)) is this total divided by the number of observations (5), resulting in 12.8. Finally, the square root of 12.8 provides a standard deviation (\( \sigma \)) of approximately 3.58.
This measure is particularly helpful in understanding how much individual data points deviate from the average value (the mean), enabling you to assess the consistency of the data.

Variance is another important statistical concept that reflects the degree of spread in a dataset. It measures how far each number in the set is from the mean and, consequently, from every other number in the set.
To compute variance, follow these steps:

• Compute the mean of the dataset.
• Subtract the mean from each number to determine the deviations.
• Square each of these deviations to mitigate the effect of direction (positive or negative values).
• Find the mean of these squared deviations.

In this exercise, we first computed the deviations from the mean (15) for each data value, then squared these deviations to get values of 25, 25, 9, 4, and 1.
Adding these squared deviations gives a total of 64, which when divided by the number of observations (5) results in a variance (\( \sigma^2 \)) of 12.8.
Variance is critical in the field of statistics as it provides the basis for the standard deviation and helps explain the variability within a dataset.