Episode 22: June 10, 2010
by Jason Marshall
In the last article, we talked about averages and one specific way statistics can help answer questions without letting emotions interfere. But the type of average value we talked about is just one of manyuseful meanings assigned to the word “average.” Today we’re going to talk about these many meanings and how to calculate the value of one of them: the arithmetic mean.
How is “Average” Defined?
As we talked about last time, batting averages in baseball tell us how often a hitter has been successful at the plate. For example, a .320 batting average means a player has hit safely in 32% of his at bats. But, besides telling us about the past successes of baseball players, what exactly is the concept of an average value good for? Well, the purpose of an average value is to find a single number to represent the typical value of an entire list of numbers.
Just what types of lists are we talking about? Well, almost anything: a list of grades on an algebra test; a list of the weights of golf balls coming off a manufacturing line; or perhaps a list of how many leaves are on each tree in an orchard. Each of these lists provides what is called a sample of data, and the calculated average value of that sample represents its typical, or average, value. So, what’s the best way to calculate this average value?
What is the Arithmetic Mean?
Well, there really isn’t a “best” way to calculate the average value—there isn’t one unique method that gives all the information we’d like to know about a sample. So then, what’s the best-known way? The value you probably most commonly associate with an average is called the arithmetic mean—usually just called the mean for short. In fact, since it’s far-and-away the most commonly used method for defining average values, it’s frequently known simply as the average. In an effort not to get our terms confused, I’m going to stick to calling it the mean. So, how is it calculated?
How to Calculate Mean Values
The purpose of an average value is to find a single number to represent the typical value of an entire list of numbers.
Imagine you have 11 fun-pack sized bags of potato chips. How many chips do you think each bag contains? Instead of guessing, let’s imagine you open each bag, count the number of chips, and carefully record those numbers on a piece of paper. I’ve never actually done this experiment, but I’m willing to bet that the bags won’t all have the same number of chips in them. After all, chips get broken and come in all kinds of funny shapes and sizes, so the number is certain to vary. After counting all the chips, let’s imagine you find that the 11 bags contain totals of 18, 15, 19, 18, 23, 17, 18, 16, 19, 34, and 17 chips. That’s your sample—it contains 11 values, one for each bag of chips.
So, what’s the mean number of chips that comes in one of these fun-pack bags?
The quick and dirty tip is that the mean value of a sample is calculated by first adding up all the numbers in the sample—in our case, the total number of chips in all the bags put together—and then dividing this total by the number of samples—in this case, that’s the number of bags. The total number of chips is 214, and since these chips are in a total of 11 bags, the mean value is 214 / 11, or about 19.45. Although it looked a little different, this is the same type of averaging used to calculate batting averages in the previous article. Check out this week’s Math Dude “Video Extra!” episode on YouTube for a closer look at this relationship, and to see a few more examples of calculating mean values.
What Does the Mean Value Mean?
But what does this mean value really mean? Well, if you think about it for a bit, you’ll find that replacing all the values in a sample with the mean value doesn’t change the sample’s total value. In other words, in our case, if there were actually 19.45 potato chips (that’s the mean value) in each of the 11 bags, we’d have a combined total of 214 chips—just as we did before. Technically, that’s exactly what the mean value means—nothing more, nothing less. It’s certainly a reasonable way to calculate a typical value for a sample, it satisfies our intuitive sense of what an average should be, and it works very well in most cases. But there’s nothing magical about it, and there are certainly other ways to calculate average values. And some of these ways are actually more useful for analyzing certain types of problems. In fact, I’d argue that one of these other ways is better suited to our problem. Why do I say that?
Well, unfortunately we’re out of time for today, so the answer to that question will have to wait until the next article about median averages. In that article, we’ll see how this type of averaging is better for certain types of data (like ours), and we’ll also talk about a pretty impressive application of median averaging that you can use to remove unwanted tourists from your vacation photos!
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!