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Finding the mean is a fundamental step in many statistical calculations, including the standard deviation. The mean provides a central point of reference for the data set. It is calculated by summing up all the observations and dividing by the number of observations. This process helps us understand the average value of the data set. In our example, we have a set of distances: 29, 37, 38, ..., 150. We add them all together to get 1716. Since there are 20 data points, we divide 1716 by 20, giving us an average, or mean, of 85.8 miles.
This mean serves as a typical value around which the other distances are spread.

Variance measures how much the data points deviate from the mean. It's crucial for understanding the dispersion in data. To calculate it, we first find how far each data point is from the mean (this is the deviation), and then we square each deviation. Squaring the deviations eliminates negative values and gives weight to larger deviations, thereby indicating how spread out the values are.
After obtaining these squared deviations, we sum them up and divide by the number of data points to get the average of the squared deviations, known as the variance. In our exercise, this step is key as it helps quantify the level of variation among store distances to the distribution center.

Deviation from the mean shows the difference between each data point and the mean value. This concept helps in identifying how far off each point is from the average. In our data set, this calculation involves subtracting the mean (85.8) from each data point.
For instance, with a first point of 29, the calculation is 29 - 85.8, resulting in a deviation of -56.8. This process is repeated for all data points, giving us an array of deviations that tell us exactly how much each distance varies around the mean. These deviations are critical for further calculations, such as variance and standard deviation.

Data analysis involves using statistical methods to interpret data sets critically. In our problem, it includes calculating the mean, deviations, variance, and standard deviation. By understanding these calculations, one can derive insights into the data's spread and central tendency.
This kind of analysis helps identify patterns or anomalies in the distances between the stores and the distribution center. It also aids in effective decision-making, as it provides quantitative measures of data spread, like the standard deviation which reflects how much individual data points differ from the mean. Such analysis is not just theoretical but practical, impacting logistics, resource allocation, and strategic planning in business contexts.