Task
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]
[Stand-alone applet]
Discussion
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.
