||Using interactive software, students can compare and contrast properties
of measures of central tendency, specifically the influence of changes
in data values on the mean and median. As students change the data
values, the interactive figure immediately displays the mean and median
of the new data set.
Experimenting with this software helps students
compare the utility of the mean and the median as measures of center for
different data sets, as discussed in the Data Analysis and Probability Standard.
The seven data points in the line plot below represent the distances a paper airplane traveled after it was thrown. Your task is to explore how changing one (or more) of the data points affects the mean and the median of the data set.
The following questions may be useful in focusing your experimentation:
- Can you find ways to move the data points that keep the median the same but change the mean?
- Can you find ways to move the data points that keep the mean the same but change the median?
- How do the mean and median change when you keep the points in the same order but just change their positions on the number line?
- What happens if you pull some of the data values way off to one extreme or the other extreme?
- By moving data points, can you construct data sets in which the mean seems to be a typical value but the median is not? Vice versa? For what types of data sets, if any, is the mean not very representative? When is the median not very representative?
How to Use the Interactive Figure
To change the value of a data point, click on the point and drag it left or right. The mean and median are updated automatically.
Mean and median are two types of "averages" or measures of central tendency. Both measures appear in everyday media reports, and they are generally studied by students in the elementary and middle grades. The median is a measure of the "middle" of the data. For an odd number of data points arranged in ascending order, the median is actually the middle value, and for an even number of data points it is the value halfway between the two middle data points. The mean (a number which "evens out" or balances a set of data) is computed by adding all the numbers in the set and dividing the sum by the number of elements added. For a given set of data, these measures of center may be very close or may be quite different, depending on how the data are distributed, and either of the measures of center may or may not provide a good measure of "typicalness."
A more visual and intuitive way of thinking about mean and median is to picture each of the values in the data set as a stack of cubes with height equal to that value. If we visualize "sharing" cubes across all the stacks to make them of equal height, that common height is the mean. To visualize the median, picture the stacks of cubes arranged from shortest to tallest. The median is the height of the middle stack, or the average of the heights of the two middle stacks if there are an even number of stacks.
The mean and median each have advantages and disadvantages when used to describe data sets. The mean depends on the actual values in a data set, but the median is dependent only on the relative position of the values. Changing one data value does not affect the median, unless the data value is moved across the middle of the data set. But every change in a data value affects the mean. Thus, the mean is affected by a few extremely large or extremely small values outside the range of the rest of the data, but the median is not.
The following questions are useful to raise in class discussion:
- What sorts of changes in a data set make the mean change?
- What sorts of changes in a data set make the median change?
Can you find examples in the popular press where the mean of a data set is cited and other examples where the median is cited? Why do you think the authors of those articles chose to cite those particular measures of center? Would readers have received a different impression of the data under discussion if other (or additional) measures of center had been reported?
Take Time to Reflect
- What kinds of questions that are of interest to students can be explored through data investigations and require the use of mean or median?
- How can involving students in data investigations help them connect mathematics with other subjects in the school curriculum?
- Measures of center are just one way that statisticians use to describe data distributions. What other statistical descriptors are useful in characterizing a set of data?