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The Median Median Line (PowerPoint)

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					The Median-Median Line

        Lesson 3.4
 Have you noticed that you and your classmates
frequently find different equations to model the
                    same data?
The median-median line is one of the
simplest methods.
The procedure for finding the median-median
line uses three points M1, M2, and M3 to
represent the entire data set, and the equation
that best fits these three points is taken as the
line of fit for the entire set of data.
To find the three points that will represent the entire data
set, you first order all the data points by their domain value
(the x-value) and then divide the data into three
equal groups.
If the number of points is not divisible by 3, then you split
them so that the first and last groups are the same size. For
example:
          18 data points: split into groups of 6-6-6
          19 data points: split into groups of 6-7-6
          20 data points: split into groups of 7-6-7
You then order the y-values within each of
the groups. The representative point for
each group has the coordinates of the
median x-value of that group and the
median y-value of that group.
Because a good line of fit divides the data
evenly, the median-median line should
pass between M1, M2, and M3, but be
closer to M1 and M3 because they
represent two-thirds of the data.
To accomplish this, you can find the y-
intercept of the line through M1 and M3,
and the y-intercept of the line through M2
that has the same slope.
The mean of the three y-intercepts of the
lines through M1, M2, and M3 gives you
the y-intercept of a line that satisfies these
requirements.
Example
 Find the median-median line
  for these data.
                  60  29
slopem1  m3                5.082
                 11.9  5.8

ym1  m3  29  5.082( x  5.8)
ym1  m3  0.475  5.082 x
                                    60  29
                  slopem1  m3                5.082
                                   11.9  5.8




yM2  44.5  5.082( x  8.6)
yM2  0.795  5.082 x
   y M2  0.795  5.082 x         ym1 m3  0.475  5.082x

The Median-Median Line
 The median-median line is parallel to both of these lines, so it
  will also have a slope of 5.082. To find the y-intercept of the
  median-median line, you find the mean of the y-intercepts of
  the lines through M1, M2, and M3 .
 Since there is one line that passes through M1 and M2, the y-
  intercept for these lines are both -0.475. The y-intercept of
  the line through M3 is 0.795.
 The mean of the y-intercepts is
             -0.475 + (-0.475) + 0.795
                                        0.052
                         3
 So the equation of the median-median line is

            ymedian  median  0.052  5.082x
    Finding a Median-Median Line
1. Order your data by domain value first. Then, divide the data into three
   sets equal in size. If the number of points does not divide evenly into
   three groups, make sure that the first and last groups are the same size.
   Find the median x-value and the median y-value in each group. Call
   these points M1, M2, and M3 .
2. Find the slope of the line through M1 and M3 . This is the slope of the
   median-median line.
3. Find the equation of the line through M1 with the slope you found in
   Step 2. The equation of the line through M3 will be the same.
4. Find the equation of the line through M2 with the slope you found in
   Step 2.
5. Find the y-intercept of the median-median line by taking the mean of
   the y-intercepts of the lines through M1, M2, and M3 . The y-intercepts
   of the lines through M1 and M3 are the same.
6. Finally, write the equation of the median-median line using the mean y-
   intercept from Step 5 and the slope from Step 2.
    Airline Schedules
 In this investigation you will
  use data about airline flights
  to find a median-median line
  to model the relationship
  between the distance of a
  flight and the flight time.You
  will use the linear model to
  make predictions about flight
  times and distances that aren’t
  in the table.
 The flights listed here are
  morning departures from
  Detroit, Michigan. Write a
  complete sentence explaining
  what the first line of data tells
  you.
 Graph the data on
  graph paper.
 Show the steps to
  calculate the
  median-median line
  through the data.
  Write the equation
  of this line. Use your   M1  (67,306)
  calculator to check      M2  (168.1033.5)
  your work.               M3  (288,1979)
                                               1979  306
                              slopeM1  M3   
                                                288  67
64    229                         1673
                                       7.57
67    306      (67, 306)           221
                             yM1  M3  306  7.57(x  67)
104   610
                             yM1  M3  201.19  7.57 x
120   658
156   938    (168, 1033.5)   yM2  1033.5  7.57( x  168)

180   1129                   yM2  238.26  7.57 x
189   1092                   yintercept 

248                          201.19  201.19  238.26
      1671                               3
                                                       
288   1979   (288, 1979)     213.55
303   2079                    ymedian  213.55  7.57x
On your graph, mark
the three
representative points
used in the median-
median process. Add
the line to this graph.
Answer these questions about your data and model.
a. Use your median-median line to interpolate two points for
which you did not collect data. What is the real-world meaning
of each of these points?
b. Which two points differ the most from the value predicted by
your equation? Explain why.
c. What is the real-world meaning of your slope?
d. Find the y-intercept of your median-median line. What is its
real-world meaning?
e. What are the domain and range for your data? Why?
f. Compare the median-median line method to the method you
used in Lesson 3.3 to find the line of fit. What are the advantages
and disadvantages of each? In your opinion, which method
produces a better line of fit? Why?

				
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