Loyola College of Maryland

Journal of Statistics Education Volume 14, Number 2 (2006), jse.amstat.org/v14n2/morrell.html

Copyright © 2006 by Elizabeth J. Walters, Christopher H. Morrell, and Richard E. Auer, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

**Key Words:** Least squares line.

The second trend is the introduction of statistical ideas earlier in the school curriculum. Younger students are being exposed to the question of what truths lurk beneath the surface of data. These students are now being trained to use statistics as the primary tool for researching any topic within any field.

Evidence of both trends is seen when elementary and secondary school students use graphing calculators to perform data analysis on exams and homework assignments. Much of the early statistical training is being accomplished by teaching the basics of what has been termed exploratory data analysis (EDA).

As the first pioneer in exploratory data analysis, Tukey (1971, p. v) effectively defined EDA as “… looking at data to see what it seems to say. It concentrates on simple arithmetic and easy-to-draw pictures. It regards whatever appearances we have recognized as partial descriptions, and tries to look beneath them for new insights.” In his recommendations for what type of activity belongs in introductory statistics coursework, Hahn (1988, p. 27) said: “When we get to data analysis, we should stress graphical methods or exploratory data analysis, over formal statistical procedures.” Cobb and Moore (1997, p. 815) echoed the sentiments of Hahn saying: “Students like exploratory analysis and find that they can do it, a substantial bonus when teaching a subject feared by many. Engaging them early on in the interpretation of results, before the harder ideas come to their attention, can help establish good habits when you get to inference.”

Similar thinking was the basis for the Quantitative Literacy program of the mid-1980’s which was sponsored by the
American Statistical Association (2005a) and the National Council of Teachers of
Mathematics Joint Committee on the Curriculum in Statistics and Probability. Funded in part by the National Science
Foundation, this program involves “… education workshops and written materials to help elementary and secondary school
teachers make statistics more accessible to students … with an emphasis on graphical techniques”
(American Statistical Association, 2005b). These written works were made
available by Dale Seymour Publications through a series of four books. All four of these books focus on a “hands-on”
approach that utilizes data very familiar to students from areas like sports, television, music, etc. One of these,
*Exploring Data* by Landwehr and Watkins (1986), introduces students to
a method of fitting a line through bivariate data using a simple application of the median.

One major goal of young statisticians as they learn exploratory data analysis is studying the relationship between two variables. When the data are bivariate with two numerical variables, this usually involves fitting a straight line through points on a scatterplot. The graphing calculator is programmed with two methods of line fitting for bivariate data. One method, fitting the least squares line, is a procedure well studied and routinely applied. But the median-median line, on the other hand, is not so well known or understood.

This paper considers the performance of these two line-fitting techniques as an aide to budding statisticians and their teachers as they encounter exploratory data analysis. We intend to illuminate the behavior of the relatively unknown median-median line with several data sets and through a small simulation study. Readers may compare how the median-median line reacts in various data settings to the performance of the least squares line.

A similar procedure suggested by Nair and Shrivastava (1942) breaks up the points
on a scatterplot into three regions with each region containing about the same number of points. The means of the *x*
and *y* points in the left and right regions are used to find the slope of the line of fit much as Wald suggested.

Brown and Mood (1951) used the two-region approach but found the slope of the line of fit using medians in place of means. The primary advantage of using this measure of center comes from the median’s inherent ability to resist the strong effect of outliers. Most students of statistics know that the mean can be affected greatly by outliers since they are included with equal weight with the rest of the data in the sum of the scores. But the median takes on the same value whether the largest score in a data set is just somewhat larger than the rest of the data or is much larger than the second biggest score. In the context of fitting points on a scatterplot, this implies that a single point far from the general sloping trend of the rest of the points would not apply such a large “tug” on the location of the line of fit if the median is used to find the line.

Readers may recall that the least squares line is found by minimizing the sum of the squared distances that each point lies from the line. Since these distances are squared, Hartwig and Dearing (1979, p. 34) noted that “… cases lying farther and farther from the regression line increase the sum of the squared residuals at an increasing rate ... [and the line] will have to come reasonably close to them to satisfy the least squares criterion and, therefore, the least squares regression line will lack resistance to the excessive influence of a few atypical cases.” This means that Brown and Mood’s method is not only simple to apply, but also has the advantage of not allowing outlying cases to have undue impact on determining the line of fit.

Like Brown and Mood, Tukey (1971) utilized the medians in finding his line of fit, but he did so borrowing the three-region approach of Nair and Shrivastava. His line of fit, called the resistant line, is considered a basic methodology of exploratory data analysis. The median-median line provides the first iteration in the procedure to find Tukey’s resistant line. To obtain Tukey’s resistant line, the residuals are used to adjust the parameters in an iterative fashion. Appendix A provides the algorithm used by Texas Instruments to compute the median-median line. Tukey’s resistant line may be obtained using Minitab under the EDA submenu of Stat.

For those who wish to learn more about the broad methodology of exploratory data analysis, Velleman and Hoaglin (1981) provide a gentle review of the entire subject. To learn more about the median-median line at a more sophisticated mathematical level, readers are encouraged to consider Emerson and Hoaglin (1983) and Johnstone and Velleman (1985).

number of powerboat registrations, in thousands, and the second column

Region 1 | Region 2 | Region 3 | ||||
---|---|---|---|---|---|---|

x | y |
x | y |
x | y | |

447 | 13 | 513 | 24 | 614 | 33 | |

460 | 21 | 526 | 15 | 645 | 39 | |

481 | 24 | 559 | 34 | 675 | 43 | |

498 | 16 | 585 | 33 | 711 | 50 | |

512 | 20 | 719 | 47 | |||

median | 482 | 20 |
542.5 | 28.5 |
675 | 43 |

Once the data are divided into three regions, the median of the *x*- and median of the *y*-scores are calculated
for each region. The resulting three points for these data are termed the median-median points
= (481, 20), = (542.5, 28.5),
= (675, 43). The slope of the median-median line is the slope of the
line passing through and ; that is,
. The position of the median-median line is determined by looking at
the line passing through and
and a parallel line passing through . Moving the line connecting the two outer points
one third of the way to the line through the point in the center region yields the median-median line (see
Figure 1). Note that the line that connects the two outer points is based on two
of the median-median points, while the line through is based on only
this median-median point. Thus, choosing to move the first line one third of the way towards the second effectively makes
the median-median line a weighted average of the outer two lines. The resulting y-intercept is
(See Appendix A. Note that
TI calls the slope *a* and the *y*-intercept *b*.). This yields the equation of the median-median line:
. Note that the median-median and least-squares lines have similar
equations, as is evident from the scatterplot (see Figure 2).

Figure 1. Illustration of the median-median method for the manatee data.

Figure 2. Scatterplot of manatees killed by powerboats versus number of boat registrations.

Figure 3. Scatterplot of Gesell Adaptive Score versus Age at Which Child Began Speaking.

Figure 4. Scatterplot of dietary energy density versus body mass index.

For illustration, we also consider DED as the explanatory variable and BMI as the response variable. In this scenario, the observation that was previously a high leverage point is now an observation with a large residual (Figure 5). Now the least squares line is not affected as much by the outlying observation as in Figure 4 where the observation has both high leverage and is far from the trend formed by the remaining observations. The data are provided in Appendix B.

Figure 5. Scatterplot of body mass index versus dietary energy density.

The conditions under which the simulation is performed are:

- two sets of values of the explanatory variable:

Set 1:*x*-values = (1, 2, 3, …, 24), and

Set 2:*x*-values = (2, 2, 4, 4, …, 24, 24), - two levels of the error standard deviation ( = 1 and 5), and
- a number of outlier possibilities.

Without loss of generality, in most cases we assume that the outlier occurs in the upper region of *X*-values, that is,
the outlier is one or two of *Y*_{17}, …, *Y*_{24}.
Table 2 provides a summary of the outliers generated in the simulation study.

Outlier Generation | Description | |
---|---|---|

a) | No Outliers | No Outliers |

b) | Y_{13} = 0 + 1X_{13}
+ 5|| |
One high outlier in the middle of the middle region |

c) | Y_{17} = 0 + 1X_{17}
+ 5||Y_{18} = 0 + 1X_{18}
- 5|| |
Outliers at the first two X-values in the upper region, one high and one low |

d) | Y_{17} = 0 + 1X_{17}
+ 5|| |
One high outlier at the first X-value in the upper region |

e) | Y_{23} = 0 + 1X_{23}
- 5||Y_{24} = 0 + 1X_{24}
+ 5|| |
Outliers at the two largest X-values, one high and one low |

f) | Y_{17} = 0 + 1X_{17}
+ 5||Y_{18} = 0 + 1X_{18}
+ 5|| |
High outliers at the first two X-values in the upper region |

g) | Y_{17} = 0 + 1X_{17}
- 5||Y_{24} = 0 + 1X_{24}
+ 5|| |
A low outliers at the first X-value in the upper region and a high outlier at the
largest X-value |

h) | Y_{24} = 0 + 1X_{24}
+ 5|| |
A single high outlier at the largest X-value |

i) | Y_{23} = 0 + 1X_{23}
+ 5||Y_{24} = 0 + 1X_{24}
+ 5|| |
High outliers at the two largest X-values |

For each type of outlier generation, 1000 samples are simulated. For each sample the least squares and median-median
estimates of the intercept and slope are computed. Tables 3 to 6 provide the
following statistics on the 1000 simulated least squares and median-median slope estimates: means, standard deviations, and
mean square errors (variance of simulated slopes plus bias squared where the bias = mean - 1) for each of the nine methods
of generating the data. If the mean of the simulated slope estimates is 1, the estimator is considered unbiased. The
standard deviation provides a measure of spread of the estimates of the slopes. And the mean square error provides a
measure of precision of how far the estimates vary from the true slope. Since the main focus of this paper is on the
slopes of the estimated line, we do not consider the estimated intercepts in this discussion of the simulation results.
These tables also show the leverage value (see Neter, Kutner, Nachtsheim, and
Wasserman (1996, p. 375-377)) for the design points at which the outlier is generated. Leverage values help to
identify outlying *X*-values that, in conjunction with extreme *Y*-observations, may lead to data points with a
large influence on the slope and intercept of the line. It is recommended to compare the leverage value with
where *p* is the number of parameters in the linear regression model. In our case *p* = 2
(for the intercept and slope) and *n* = 24 so = 0.167. In our
example, no design point has a leverage that exceeds this value though the leverage of the most extreme *X*-values
(1 and 24) come close to .

Model: *Y* = 0 + 1*X* + ; X’s: Set 1;
~ N(0, 1^{2}).

Lower region: *X*_{1}, …, *X*_{8};
Middle region: *X*_{9}, …, *X*_{16};
Upper region:*X*_{17}, …, *X*_{24}.

Mean Slope Estimate | Standard Deviation | MSE (x 10^{3}) | |||||
---|---|---|---|---|---|---|---|

Type of Outlier | Leverage | LS | MM | LS | MM | LS | MM |

a) No outliers | 1.0017 | 1.0037 | 0.0295 | 0.0509 | 0.873 | 2.604 | |

b) One high in middle of middle region | 0.0419 | 1.0035 | 1.0037 | 0.0295 | 0.0509 | 0.882 | 2.604 |

c) First two X-values in upper region, one high and one low | 0.0593 0.0680 |
0.9980 | 1.0291 | 0.0349 | 0.0568 | 1.222 | 4.073 |

d) One high at first X-value in upper region | 0.0593 | 1.0170 | 1.0295 | 0.0315 | 0.0564 | 1.285 | 4.051 |

e) Two largest X-values, one high and one low | 0.1375 0.1567 |
1.0054 | 0.9703 | 0.0472 | 0.0557 | 2.257 | 3.985 |

f) High at first two X-values in upper region | 0.0593 0.0680 |
1.0357 | 1.0571 | 0.0338 | 0.0571 | 2.417 | 6.521 |

g) Low at first X-value in upper region, high at largest X-value | 0.0593 0.1567 |
1.0254 | 1.0037 | 0.0424 | 0.0511 | 2.442 | 2.625 |

h) High at largest X-value | 0.1567 | 1.0405 | 1.0038 | 0.0405 | 0.0510 | 3.280 | 2.615 |

i) High at two largest X-values | 0.1375 0.1567 |
1.0749 | 1.0047 | 0.0482 | 0.0513 | 7.933 | 2.654 |

Model: *Y* = 0 + 1*X* + ; *X*’s:
Set 1; ~ N(0, 5^{2}).

Lower region: *X*_{1}, …, *X*_{8};
Middle region: *X*_{9}, …, *X*_{16};
Upper region:*X*_{17}, …, *X*_{24}.

Mean Slope Estimate | Standard Deviation | MSE (x 10^{3}) | |||||
---|---|---|---|---|---|---|---|

Type of Outlier | Leverage | LS | MM | LS | MM | LS | MM |

a) No outliers | 1.0087 | 1.0141 | 0.1473 | 0.1850 | 21.773 | 34.424 | |

b) One high in middle of middle region | 0.0419 | 1.0173 | 1.0141 | 0.1475 | 0.1850 | 22.056 | 34.424 |

c) First two X-values in upper region, one high and one low | 0.0593 0.0680 |
0.9902 | 1.0614 | 0.1746 | 0.1948 | 30.581 | 41.717 |

d) One high at first X-value in upper region | 0.0593 | 1.0848 | 1.0895 | 0.1576 | 0.1898 | 32.029 | 44.034 |

e) Two largest X-values, one high and one low | 0.1375 0.1567 |
1.0272 | 0.9619 | 0.2362 | 0.1985 | 56.530 | 40.854 |

f) High at first two X-values in upper region | 0.0593 0.0680 |
1.1786 | 1.1702 | 0.1691 | 0.1952 | 60.493 | 67.071 |

g) Low at first X-value in upper region, high at largest X-value | 0.0593 0.1567 |
1.1271 | 1.0155 | 0.2118 | 0.1969 | 61.014 | 39.010 |

h) High at largest X-value | 0.1567 | 1.2023 | 1.0423 | 0.2026 | 0.1913 | 81.972 | 38.385 |

i) High at two largest X-values | 0.1375 0.1567 |
1.3743 | 1.0888 | 0.2408 | 0.2009 | 198.085 | 48.246 |

Model: *Y* = 0 + 1*X* + ; *X*’s:
Set 2; ~ N(0, 1^{2}).

Lower region: *X*_{1}, …, *X*_{8};
Middle region: *X*_{9}, …, *X*_{16};
Upper region:*X*_{17}, …, *X*_{24}.

Mean Slope Estimate | Standard Deviation | MSE (x 10^{3}) | |||||
---|---|---|---|---|---|---|---|

Type of Outlier | Leverage | LS | MM | LS | MM | LS | MM |

a) No outliers | 1.0018 | 1.0040 | 0.0296 | 0.0540 | 0.879 | 2.556 | |

b) One high in middle of middle region | 0.0425 | 1.0053 | 1.0040 | 0.0297 | 0.0504 | 0.910 | 2.556 |

c) First two X-values in upper region, one high and one low | 0.0635 0.0635 |
1.0015 | 1.0342 | 0.0350 | 0.0569 | 1.227 | 4.407 |

d) One high at first X-value in upper region | 0.0635 | 1.0188 | 1.0343 | 0.0321 | 0.0567 | 1.384 | 4.391 |

e) Two largest X-values, one high and one low | 0.1474 0.1474 |
1.0022 | 0.9741 | 0.0474 | 0.0553 | 2.252 | 3.729 |

f) High at first two X-values in upper region | 0.0635 0.0635 |
1.0359 | 1.0573 | 0.0340 | 0.0572 | 2.445 | 6.555 |

g) Low at first X-value in upper region, high at largest X-value | 0.0635 0.1474 |
1.0222 | 1.0039 | 0.0423 | 0.0509 | 2.282 | 2.606 |

h) High at largest X-value | 0.1474 | 1.0390 | 1.0042 | 0.0400 | 0.0506 | 3.121 | 2.578 |

i) High at two largest X-values | 0.1474 0.1474 |
1.0753 | 1.0047 | 0.0485 | 0.0507 | 8.022 | 2.593 |

Model: Y = 0 + 1X + ; *X*’s:
Set 2; ~ N(0, 5^{2}).

*X*_{1}, …, *X*_{8};
Middle region: *X*_{9}, …, *X*_{16};
Upper region:*X*_{17}, …, *X*_{24}.

Mean Slope Estimate | Standard Deviation | MSE (x 10^{3}) | |||||
---|---|---|---|---|---|---|---|

Type of Outlier | Leverage | LS | MM | LS | MM | LS | MM |

a) No outliers | 1.0090 | 1.0138 | 0.1479 | 0.1849 | 21.955 | 34.378 | |

b) One high in middle of middle region | 0.0425 | 1.0263 | 1.0138 | 0.1485 | 0.1849 | 22.744 | 34.378 |

c) First two X-values in upper region, one high and one low | 0.0635 0.0635 |
1.0076 | 1.0628 | 0.1751 | 0.1949 | 30.718 | 41.930 |

d) One high at first X-value in upper region | 0.0635 | 1.0940 | 1.0876 | 0.1607 | 0.1898 | 34.660 | 44.698 |

e) Two largest X-values, one high and one low | 0.1474 0.1474 |
1.0108 | 0.9632 | 0.2371 | 0.1988 | 56.333 | 40.874 |

f) High at first two X-values in upper region | 0.0635 0.0635 |
1.1797 | 1.1700 | 0.1700 | 0.1942 | 61.192 | 66.614 |

g) Low at first X-value in upper region, high at largest X-value | 0.0635 0.1474 |
1.1112 | 1.0154 | 0.2114 | 0.1980 | 57.055 | 39.441 |

h) High at largest X-value | 0.1474 | 1.1952 | 1.0467 | 0.2001 | 0.1917 | 78.143 | 38.930 |

i) High at two largest X-values | 0.1474 0.1474 |
1.3763 | 1.0886 | 0.2427 | 0.2008 | 200.505 | 48.171 |

Figures 6 to 8 provide density estimates for three of these examples for the first
set of *x*-values (1, 2, 3, …, 24): no outliers, a moderate outlier example (high and low outliers at the end –
Table 2, design (e)), and the most extreme outlier example (two high outliers at
the end – Table 2, design (i)). The density estimates are simply the simulated
distributions of the slope estimates and are computed from the 1000 slope estimates using the SPlus function density.

The simulation study comparing the performance of the two methods of regression when no outliers are present (Figure 6) shows that both the median-median and LS estimates are close to being unbiased. However, there is less variation in the least-squares slope estimates than in the median-median slope estimates.

Figure 6. Density estimates of the slopes for the 1000 replications for least squares and median-median estimates and for
= 1 and = 5;

*X*’s: Set 1; no outliers.

In the moderate case (Figure 7, one high and one low outlier at the end – Table 2, design (e)), the least-squares method still tends to give unbiased estimates of the slope while the median-median estimates are slightly biased to the low side and exhibit smaller MSE than the LS line when = 5.

Figure 7. Density estimates of the slopes for the 1000 replications for least squares and median-median estimates and for
= 1 and = 5;

*X*’s: Set 1; one high and one low outlier at the end – Table 2, design (e).

In the extreme case (Figure 8, two high outliers at the end – Table 2, design (i)), however, the median-median method performs much better, with less variation and more accuracy than the least-squares method, especially in the case of greater population standard deviation ( = 5 vs. = 1).

Figure 8. Density estimates of the slopes for the 1000 replications for least squares and median-median estimates and for
= 1 and = 5;

*X*’s: Set 1; two high outliers at the end – Table 2, design (i).

The three examples provided in this paper show that, when an influential point is present, the median-median line may be
less affected by the influential point than the least-squares line. The simulation study further supports the argument
for using the median-median line as a resistant method of regression when outliers are present, especially in the most
extreme cases (one or two high outliers at the end; see Table 2, designs (h and i))
and in cases with greater population variance when the outlier is on the most extreme *x*-value.

On the other hand, there exists a well-developed set of tools for inference and for detecting the presence of unusual observations when using least-squares regression under normal error assumptions. In contrast, inference about the median-median estimates would require bootstrapping to obtain approximate standard errors, confidence intervals/regions, and p-values for tests. Given this, in addition to the superior performance of least-squares estimation when either no outliers or moderate outliers are present, the median-median method of regression would not appear to be a valuable tool at either the professional or collegiate level.

However, in elementary and middle schools the median-median method of estimating a line may be a reasonable approach to describe a linear relationship in a scatterplot. The teacher may first introduce the idea of a scatterplot as a way of visualizing the association between a response and explanatory variable. If there is a straight-line trend in the plot, the straight line may be fit by eye using a ruler to try to capture the trend. While each student will likely be satisfied that their eye-balled line fits the data well, they will also recognize that each student would have chosen a slightly different line. This will suggest the need for a more objective and structured method of fitting a line. The median-median line will not only be simple to find, but may also feel very connected to the visual process each student had just undertaken.

Median-Median Line Algorithm on Graphing Handhelds. Solution: What method is used to calculate the median-median line? The goal of the median-median line is to: 1) Divide the data into three parts with an equal number of data points 2) Define a summary point for each part 3) Use the three summary points to define the median-median line How the TI Calculator does this: 1) The calculator will attempt to break the list into three equal parts without breaking up data groups of equal x values. In this case, the algorithm we use is designed to include at least 1/3 of the points in the left and right groups. What ever is left over is put into the center group, hopefully the remaining 1/3. The approach chosen was to fill the outside groups first and allocate the remaining data points to the center group. If the center group is empty, an error message is generated. The reason equally x values are not split up is to ensure the same results are produced independent of the order the data appears in the original list. Without this restriction, a different result could be produced depending on the ordering in the data input lists. 2) A summary point is simply the median of all x's and y's in that part. Let's call the summary points (x1,y1), (x2,y2), and (x3,y3). 3) The median-median line will be parallel to the line going through the points (x1,y1) and (x3,y3) a = (y3 - y1) / (x3 - x1) and 1/3 the distance between the line through the two summary points and a parallel line going through the second summary point (x2,y2). b = (y1 + y2 + y3 - a (x1 + x2 + x3)) / 3. An advantage of the median-median line over a least-squares line, is that stray data points do not affect the end result very much.

Power Boat Registrations | Manatee Deaths |
---|---|

447 | 13 |

460 | 21 |

481 | 24 |

498 | 16 |

512 | 20 |

513 | 24 |

526 | 15 |

559 | 34 |

585 | 33 |

614 | 33 |

645 | 39 |

675 | 43 |

711 | 50 |

719 | 47 |

**Example 2**. Gessell Scores and Age illustrating the 3 regions (data used with permission of W.H. Freeman and Company)

Age (x) | Gessell Score (y) |
---|---|

7 | 113 |

8 | 104 |

9 | 91 |

9 | 96 |

10 | 83 |

10 | 83 |

10 | 100 |

10 | 100 |

11 | 100 |

11 | 84 |

11 | 102 |

11 | 86 |

12 | 105 |

15 | 95 |

15 | 102 |

17 | 121 |

18 | 93 |

20 | 87 |

20 | 94 |

26 | 71 |

42 | 57 |

**Example 3**. Body Mass Index (BMI) and Dietary Energy Density (DED) illustrating the 3 regions (data used with
permission of Thompson Learning)

BMI (x) | DED (y) | DED (x) | BMI (y) | |
---|---|---|---|---|

21.1 | 0.54 | 0.44 | 21.5 | |

21.5 | 0.44 | 0.54 | 21.1 | |

22.1 | 0.67 | 0.67 | 22.1 | |

22.3 | 0.78 | 0.78 | 22.3 | |

22.4 | 0.90 | 0.86 | 22.8 | |

22.8 | 0.86 | 0.90 | 22.4 | |

23.1 | 0.91 | 0.91 | 23.1 | |

23.3 | 0.94 | 0.93 | 26.8 | |

26.8 | 0.93 | 0.94 | 23.3 | |

From *Statistics: The Exploration and Analysis of Data* (with CD-ROM), 4^{th} Ed., by Devore/Peck, 2001.
Reprinted with permission of Brooks/Cole, a division of Thompson Learning:
www.thomsonrights.com. Fax 800 730-2215.

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Elizabeth J. Walters

Mathematical Sciences Department

Loyola College in Maryland

Baltimore, MD 21210-2699

U.S.A.
*ewalters@loyola.edu*

Christopher H. Morrell

Mathematical Sciences Department

Loyola College in Maryland

Baltimore, MD 21210-2699

U.S.A.
*chm@loyola.edu*

Richard E. Auer

Mathematical Sciences Department

Loyola College in Maryland

Baltimore, MD 21210-2699

U.S.A.
*rea@loyola.edu*

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