All Terrain Thinking

A Compendium of things I think are Important

Statistics: It's not just whats' in your wallet

What's the average? We have heard the question many times: sports fans may ask about batting averages, travelers will ask about on-time averages, university administrators will ask about retention rates (average number of students that return after their first year), investors will ask about average rates of return... But how do we go about answering the questions, computing the averages? The first thing to realize is that the average is a measure of central tendency. In this section we will talk about three measures of central tendency, the mean, median, and mode. To better understand the difference between the three measures, let's return to our example.

Mean: This is what people generally mean when they say average. The mean is the arithmetic mean of all the data which is defined as the sum of all possible values divided by the number of observations. In the grade example, we would divide the sum of all the grades (1109.08) by the number of students (52) to get the average (84.54).

Median: The median is the midpoint in the distribution of grades-is a number which is greater than half the data set and smaller than half the data set. At the median score there are as many people above the score as there are below it. The easiest way to do this would be to sort the data by score and pick the midpoint (if it is an even number of observation just average the two middle values).

Mode: The mode refers to the most frequently observed number. If we look at the scores and round them to whole numbers and sort them we can find that the number 77 appears four times and the number 92 appears 5 times. These would be the modes in the distribution.

In this example we have looked at three measures of central tendency, each of which gives us a bit of information on the class' performance on the exam. The fact that they are all different provides the sophisticated observer with additional information. For example, if the mean, median, and mode are the same, you are most likely looking at a symmetric distribution. If the mean tends to be above the median, then you probably have an outlier to the right which would give you a long upper tail which would translate into a skew to the right in the distribution. This is what we would be likely to see in income statistics if there were a few very wealthy people in the sample. Now it is time to move on to a discussion of variability.

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