The standard convention for computing the median for a dataset is the following sequence:
2a. If the size of the set is odd, choose the middle value.
2b. If the size of the set is even, average the middle pair of values.
The step 2b insures that we have a unique value for each dataset. However, if we think of the median as the number fulfilling a particular goal, such as “split the data in half,” the number may not be unique. For each perspective in this Demonstration, the choice of the median achieves a particular goal.
Snapshot 1. Splitting:
Circles on the number line represent data values. We wish to split the data into two equally sized sets. Slide
so that the number of data points above the median and the number below the median are equal. Is this possible in dataset 2? Why or why not? How can you modify the criteria to work for all datasets?
Snapshot 2: Data Reduction.
Find the value closest to the data. The upper graph shows the sum of the absolute deviations from .
In the lower graph, circles on the number line represent data values. Each red line represents the distance from an observation to
. The thickness of the line is proportional to the number of data points at that distance. Choose
to minimize the sum of absolute deviations. Note this is equivalent to minimizing the mean absolute deviation. Is there a unique minimizer for dataset 3? Why or why not?
Snapshot 3: Balance Point.
The circles represent data values on a number line that has been tied into a loop and placed on a pulley Lynch(2009). The diameter (weight) of the circle is proportional to the number of data points at that value. If the loop is rotated and released, the loop will adjust until a median is at the lowest point of the loop. Slide
to a value, click the Play trigger and see the loop move into equilibrium. Reset (with the third button) and try again with another value of
. What happens when
is far from the median? For dataset 3, what happens when
is between the two middle ordered values?
Snapshot 4: Projection.
If we use distance functions other than the Euclidean distance, we lose the concept of orthogonality. However, we can still define a projection of the data vector
onto subspace by minimizing the distance. In this case, we will use the taxicab or
distance, which is equal to sum of the absolute differences in the coordinates. Since the concept of orthogonality is not valid, a parallel plot of the vector space can be a useful way to visualize higher dimensions. In a parallel plot, a vector is represented by line segments connecting values on parallel number lines that represent the axes. The median is determined by the
. To find the projection, choose
so that the distance
is as small as possible. Compare this to the Data Reduction: least absolute deviations (LAD) perspective. Is there a unique projection for dataset 3? Why or why not?
Mark Lynch, “Teaching Tip: The Median is a Balance Point,” The College Mathematics Journal
(4), 2009 p. 292.