Measures of Center

Learning Objectives

• Find the mean, median, and mode of a set of numbers.
• Find the range and midrange of a data set.
• Read and interpret data from box-and-whisker plots.
• Solve application problems that require the calculation of the mean, median, range, or midrange.

MeanThe sum of all the data values in a data set divided by the number of items in the data set; also called the average., medianThe middle number or the mean of the two middle numbers of a set of ordered data., and modeThe number that appears most often in a data set. are important tools in the statistician’s toolbox. These measures of center all use data points to approximate and understand a “middle value” or “average” of a given data set. Two more measures of interest are the rangeThe set of all possible outputs in a function. Also the difference between the greatest value of a data set and the least value. and midrangeThe mean of the greatest and least values of a data set., which use the greatest and least values of the data set to help describe the spread of the data.

So why would you need to find out the middle of a data set? And why do you need three measures instead of just one? Let’s look closely at these measures of center and learn how they can help us understand sets of data.

“Mean” is a mathematical term for “average” which you may already know. Also referred to as the “arithmetic mean,” it is found by adding together all the data values in a set and dividing that sum by the number of data items.

You can often find the average of two familiar numbers, such as `10` and `16`, in your head without much calculation. What number lies half way between them? `13`. A mathematical way to solve this, though, is to add `10` and `16` (which gives you `26`) and then divide by `2` (since there are `2` numbers in the data set). `26-:2=13`

Knowing the process helps when you need to find the mean of more than two numbers. For example, if you are asked to find the mean of the numbers `2, 5, 3, 4, 5`, and `5`, first find the sum: `2 + 5 + 3 + 4 + 5 + 5 = 24`. Then, divide this sum by the number of numbers in the set, which is `6`. So the mean of the data is `24-:6`, or `4`.

In the previous data set, notice that the mean was `4` and that the set also contained a value of `4`. This does not always occur. In the example that follows, the mean is `18`, although `18` is not in the data set at all.

Next, let’s look at the “median.” The median is the middle value when the data is ordered. If there are two middle values, the median is the average of the two middle values.

To calculate the median, you first put your data into numerical order from least to greatest. Then identify the middle value(s).

For example, let’s look at the following values: `4, 5, 1, 3, 2, 7, 6`. To find the median of this set, you would put it in order from least to greatest.

`1text( )2text( )3text( )bb4text( )5text( )6text( )7`

Then identify the middle value. There are three values to the right of four and three values to the left of four. The middle value is `4`, so `4` is the median.

If there is an even number of data items, however, the median will be the mean of the two center data items.

Finally, let’s consider the “mode.” The mode is found by looking for the data value that appears most often. If there is a two-way tie for most often, the data is bimodal and you use both data values as the modes. Sometimes there is no mode. This happens when there is no data value that occurs most often. In our example data set (`2,3,4,bb5,bb5,bb5`), the number `5` appears `3` times and all other numbers appear once, so the mode is `5`.

Let’s look at an example with some relevant data.

What can be learned from the mean, median, and mode of Carlos’ test scores? Notice that these values are not the same.

Both the mean and the median give us a picture of how Carlos is doing. Looking at these measures, you notice that the middle of the data set is in the mid-`80`s: the mean value is `86`, and the median value is `84`. That’s all you are really after when using median and mean: finding the center, or middle, of the data. Notice, also, that there is no mode, since Carlos did not score the same on two tests. In the case of test taking, the mode is often meaningless, unless there are a lot of `0`s, which could mean that the student didn’t do his homework, or really doesn’t know what’s going on!

In this case, the mean, median, and mode are very close in value. This shows some consistency in the data, with a middle (average) value of about `11`. If this data represented the ages of students on a chess team, for example, you would have a good idea that everyone on the team was about `11` years old, with a few older and younger members.

There are other useful measures other than mean, median, and mode to help you analyze a data set. When looking at data, you often want to understand the spread of the data: the gap between the greatest number and the least number. This is the range of the data. To find the range, subtract the least value of the data set from the greatest value. For example, in the data of `2, 5, 3, 4, 5`, and `5`, the least value is `2` and the greatest value is `5`, so the range is `5 - 2`, or `3`.

It is also useful to know what number is mid-way between the least value and the greatest value of the data set. This number is called the midrange. To find the midrange, add together the least and greatest values and divide by two, or in other words, find the mean of the least and greatest values.

The midrange of `2, 5, 3, 4, 5`, and `5` is `(2+5)/2=7/2=3.5`.

Let’s look at a couple of examples.

Another type of graph that you might see is called a box-and-whisker plotA graph that uses a number line to show the distribution of a set of data.. These graphs provide a visual way of understanding both the range and the middle of a data set.

Here is a sample set of `15` numbers to get us started.

`12, 5, 18,20, 11, 9, 3, 5, 7, 18, 12, 15, 6, 10, 11`

Creating a box-and-whisker plot from this data requires finding the median of the set. To do this, order the data.

`3, 5, 5, 6, 7, 9, 10, bb11, 11, 12, 12, 15, 18, 18, 20`

This data set has `15` numbers, so the median will be the `8text(th)` number in the set: `11`.

Finding the median of the data set essentially divides it into two: a set of numbers below the median, and a set of numbers above the median. A box-and-whisker plot requires you to find the median of these numbers as well!

Lower set: `3,5,5,6,7,9,10`. Median: `6`

Upper set: `11,12,12,15,18,18,20`. Median: `15`

So, the median of the set is `11`, the median of the lower half is `6`, and the median of the upper half is `15`.

`3, 5, 5, bb6, 7, 9, 10, bb11, 11, 12,12,bb15, 18, 18, 20`

A box-and-whisker plot for this data set is shown here. Do you notice any similarities between the numbers above and the location of the box?

Notice that one “box” (rectangle section) begins at `6` (the median of the lower set) and goes to `11` (the median of the full set), and the other box goes from `11` to `15` (the median of the upper set).

The “whiskers” are the line segments on either end. One stretches from `3` (the least value in the set) to `6`, and the other goes from `15` to `20` (the greatest value in the set).

The box-and-whisker plot essentially divides the data set into four sections (or quartilesThe name of quarter sections of an ordered set of data.): whisker, box, box, whisker. The size of the quartiles may be different, but the number of data points in each quartile is the same.

You can use a box-and-whisker plot to analyze how data in a set are distributed. You can also use the box-and-whisker plots to compare two sets of data.

Using mean, median, and mode, as well as range and midrange can help you to analyze situations and make decisions about things like which is the best, whether it is more reliable to walk or take the bus to school, or even whether to buy or sell a particular stock on the stock market.

Let’s look at an example of how analyzing data using measures of center can help you to make choices (and even get to school on time!).

In the previous example, riding the bus is, on average, a faster way to school than walking. This is revealed in the mean of each method, which shows that the bus is `3` minutes faster. The mode and median show an even greater time advantage to riding the bus, and this is due to the one time high value of `31` minutes that isn’t really accounted for in these measures. Notice the difference in the mean (`17.5`) and the median (`15`) for riding the bus, which lets you know there is some variance in the data.

As far as getting to school on time is concerned, while not being the fastest method, walking is the most reliable, with consistent values for mean, median, and mode, and a low value for the range, meaning that the spread of the data is very small.

Let’s look at another example.

Depending on your viewpoint (and your preference of player!), you could make an argument that either Federer or Nadal is better than the other, but the data seems to show that they are nearly equal. Clearly, over time, the performance of Nadal and Federer is very close, with the mean of their rankings identical. The mean probably gives the best evaluation of overall performance, but does not tell us the whole story. Using the range, Nadal has been more consistent than Federer, although he has fewer number one rankings. Not shown in any of these measures is how Djokovic’s ranking has been moving upward and Federer’s ranking has been moving downward. It is also worth noting that it is hard to make a conclusion with such a small amount of data; using other measures, like tournaments won and average ranking of opponents may also help you understand who the best player was over this time.

Measures of center help you to analyze numerical data. The mean (or arithmetic mean) is often called the “average,” and is found by dividing the sum of the data items by the number of items. The median is the number that is in the middle when the data is ordered from least to greatest, and the mode is the number that appears most often. The range is the difference between the least number and the greatest number, and the midrange is the mean of the greatest value and the least value. Box-and-whisker plots use the median and range to help you to interpret the data visually.