The middle of the data: Means and medians

- Every new set of data is filled with mystery. You have no idea what it contains. Will it tell us something odd, something interesting, or will it just confirm something we already knew? So when given a new set of data, where should we get started? How can we begin to feel comfortable with our new data set? Again, let's remember our data set is a collection of values. Some might be big, some might be small and individually these values are likely too much to handle, but we're hoping together all of these values will tell us a story.

So what might be a good beginning to this story? Oddly enough for many, they like to start their story in the middle. With so many data points, wouldn't it be nice to know the center of the data? It makes sense. Knowing the center of the data would seem to give us some balance. The bigger question is what do we mean by the center of the data? For many, the center of the data would be the average, also called the mean.

This is the sum of all the data points divided by the total number of observations. Looking at data set one, our test scores, we can add our 25 test scores and then divide by 25. Our average is 65%. That doesn't look like a very good average, does it? The students in the class might complain that the exam was too difficult. That would be one way to look at those results. Another common way to find the center of the data though is by finding the midpoint.

We call this the median. For this, we organize our 25 exam scores from top to bottom. With 25 values, the 13th value is our midpoint or median. Why? The 13th value has 12 values above and 12 values below it. When we look at our exam data this way, we find that the median student's score is a 76. This would seem to indicate that the exam might have been quite fair for those that studied.

The problem might have been for those that did not study. They were doomed to get horrible test scores and when we look at the lowest test scores, we can see that just a few students really brought down the course average. Remember though, there could be so many different explanations for these outcomes. Perhaps some students were not qualified to take this course. Maybe a majority of the class had heard the exams were going to be extremely difficult and hired tutors.

Perhaps the lowest scores were earned by students that could not afford tutors. How about if many of the lowest performing students were exchange students that struggled with language and reading? As you can see, the mean and median help us identify two different types of centers. When we investigate the exam scores, the median and average scores each tell us a different story. Neither is complete, but together they have helped write the beginning of our story.

Governments, businesses, and the media love to provide people with means and medians. Often they are intended to tell you a story. Next time you are given a mean or a median, don't look at those numbers as the end of a story. Look at them as the beginning of an adventure or mystery. Start to figure out how they might provide clues. Start to imagine how the individual data points in your data set might look. Come up with questions you'd like to ask.

You never know, it's possible that their mean and medians aren't telling a true story. Perhaps they are just part of someone's fantasy.