Calculate (a) the mean and (b) the standard deviation of the data set $$ 17,26,31,19,25,20,19,29,11,27 $$

The mean, often referred to as the average, is a measure that represents the typical value of a set of numbers. It is calculated by adding together all the values in a data set and then dividing the result by the total number of points in the set.

For example, if we are given a data set such as \( 17, 26, 31, 19, 25, 20, 19, 29, 11, 27 \), the sum of these numbers is \( 224 \). With 10 numbers in this data set, we divide the sum by 10, resulting in a mean of \( 22.4 \). The mean gives us a central value which can be useful in understanding the overall distribution of data points.

Variance is a statistical measure that tells us how much the individual data points in a set vary from the mean. To calculate the variance, we first need to determine each data point's deviation from the mean by subtracting the mean from the data point value. Each of these deviations is then squared to remove negative values.

The formula for variance \( s^2 \) is given by the sum of squared deviations divided by the total number of data points minus one (in cases where we calculate sample variance) or just by the number of data points (for population variance). In our example, the variance is calculated as \( \frac{346.68}{10} = 34.668 \) by dividing the sum of squared deviations by the data points.

Squared deviations are used in the process of calculating variance and involve squaring the difference between each data point and the mean. This step is crucial because it achieves two things: it makes all the differences positive (as squaring a negative number results in a positive) and it gives more weight to larger deviations.

In our data set, for a data point like 17, the deviation from the mean (22.4) is \( -5.4 \), and its squared deviation is \( (-5.4)^2 = 29.16 \). Calculating the squared deviations for each data point and summing them up is vital for the next step in calculating variance and standard deviation.

Analyzing a data set involves using statistical measures like the mean and standard deviation to describe its characteristics. The mean provides a central value, while the standard deviation tells us about the spread of the data. In other words, standard deviation helps us understand how concentrated or spread out the data points are around the mean.

A small standard deviation suggests that the data points are closely clustered around the mean, whereas a high standard deviation indicates that the data points are spread out over a wider range of values. For the provided data set, with a mean of 22.4 and a standard deviation of approximately 5.89, one can conclude that while there are variations in the data, they are not extremely wide since the standard deviation is relatively small compared to the mean.