30 FREQUENCY DISTRIBUTIONS AND VISUAL DISPLAYS OF DATA W í fôMi ä ä 3.12 Misleading Graphs: Haw to Lie with Statistics 50- i i i i i i (3 -í u tfi "n o c 30-auf i i i < i ,»*■ ■ / \ 'im "ii i • ' 85 fiľ- 20 i í » 1 1 1 1 1 Ti i i i i i i i i_ i ____i _ j U QJ 01 > i i i i i i U i i i < i i i i i i 1900 1910 1920 1930 19*0 1950 1960 1970 1980 1990 2000 Year 31 FIGURE 3.11 An Illustration of a Time Series Graph, The Demise of Free Enterprise" from Augustine (1978). TIME-SERIES GRAPHS ■^The time-series graph, a standard statistical figure in business and economics, is becoming /common in some areas of education and psychology. It can be useful for identifying trends 'und changes in trends in ways that other representations of data cannot. A time-series graph £i8:'a fine in which the X-axis, or baseline,! is time and the vertical axis is a measure of the ^variable of interest. The time dimension can be measured in minutes, hours, days, weeks, months, or years, depending on the view tri at one wishes to take. Familiar examples of tiroe-iséries graphs include the Dow-Jones stock price average plotted across days, the Consumer SBrice Index plotted across months, a patient's body temperature plotted across hours, and :;jschbol enrollment plotted across years. Figure 3.11 is an illustrative time-series graph, "The Remise of Free Enterprise," provided byj Augustíne (1978). The extrapolated projections iihtothe future are shown by the dashed line. Obviously, such projections into the distant Ifjiture may have a large margin of error. ESS MISLEADING GRAPHS: HOW TO LIE WITH STATISTICS15 |l®£:#h'lity to interpret properly, and not be misled by, information that is presented ^■graphically is an important type of literacy for the both the layman and the professional. ||jpie-: general public is continually bombarded with data-based figures in newspapers and l^rinlly, something practical that you can usol 32 3 FREQUENCY DISTRIBUTIONS AND VISUAL DISPLAYS OF DATA magazines. Textbooks in all empirical disciplines are filled with graphs. Standardized achievement tests and university entrance tests are heavily weighted with graphic information that must be read critically (Tufte, 1983). Just as words can be misused to obscure the facts, so can pictures. At times, self-interest tempts one (including researchers) to use literal facts in such a way that the message is distorted. This may not be lying in a legal sense, but it accomplishes the same purpose. Graphs and charts can be organized so that they become propaganda rather than to illuminate the truth. Many, if not most, Figures in the popular media are constructed to be as remarkable ("newsworthy") as possible; journalists are trained to tell an interesting story, regardless of whether words or pictures are used to tell the story. It behooves us to be on our toes so we are not credulous victims of misinformation.16 Distorted Representation A common, but not very subtle error, evident in many pictographs17 found in the popular media is the linear-area fallacy. To get "more bang for the buck," graphic artists often repre- F1GURE3.12A Misleading Graphs: Illustrations of Distorted Representation.,s lfiSeveral of the examples are from ilie excellent resource book by Tufte (1983). '^Hisiognims lhal use figures lo represent frequencies. "Sources for Figures. 3.I2A: Washington Post, Oct. 25,1978, p. I; Figure 3.12B: New York Times, Aug. 9. I97B, p. D-2; Figure 3.12C: Rocky Mountain News. Muy 8,1994. p. 81 A. 3.12 Misleading Graphs: How to Lie with Statistics 33 t H /SSs^ 'i íSfts* ■ ■ .III-. -■■Vs " I Economy Standards for Autos ly Congress and the Transportalion irtment {miles per gallon) /2 FIGURE 3.12B | Misleading Graphs: Illustrations of Distorted Representation. .1 ' sent the frequency in a category by the height of the figure (a linear distance), yet make only one figure per category. This lack of uniform representation of a frequency conveys a distorted picture of the data. Notice in Figure! 3.12A how the amount of inflation is exaggerated across die five presidencies. The data are scaled by the lengrh of the dollar bill, but it is the aiea of the bills that the reader perceives. The area of the Carter dollar is less than 20% that of the Eisenhower bill, whereas the proper comparison is 100 to 44, not 100 to 20. If the bills were of the same width as the Eisenhower dollar, but were fragments with different lengths, the representation would be fair imd accurate. (Isn't the result dramatic enough without fudging7) The same flaw is seen in Figure 3.12B. Figure 3.12C gives a hopelessly distorted picture of the data.19 Misleading Scaling and Calibration There is no obvious calibration in Figure 3.12C. A more common shortcoming of graphs ..appears in Figure 3.13, where an arbitrary beginning scale value on the vertical axis is used. Variables that represent ratio scales should begin with zero to give a proper perspective for the visual interpretation. Figure 3.13 A is a common method of perceptual exaggeration; the change o\ er time is made to appear much larger than it is. Notice how different the magnitude of the change appears in a proper figure like Figure 3.13B. Many graphs (e.g., stock prices) typically ignore the zero point ana thus perceptually exaggerate the magnitude of ^changes ' Combination Graphs r"mbination graphs can be one of the most devious ways of giving unwarranted credibility aphic propaganda. All three graphs in Figure 3.14 use the same data, but note that the nd middle graphs lead to opposite conclusions! This is possible by an inappropriate ng of bath variables. Combination graphs need to be scrutinized (Wainer, 1992); the il information is usually much less convincing than the graph. Caveat emptor! Why not construct u Mr histoiiram lo gel the facts mraiglu? Use "Year" as the baseline, and MPG as ilia ul axn 34 FREQUENCY DISTRIBUTIONS AND VISUAL DISPLAYS OF DATA Denver's Number Two Paper Rooky —n Newa 1,503,962 '' __________ =^^^^^^atlo„s of DM»«. ■*»»-»• The lower graph in Figure 3.14 is fair, but still equivocal. How does one properly set the SAT? Lilce all cognitive and affective measures, it has no meaningful zero point becau it is not a ratio scale. Since 400 is the lowest possible score and 1600 the highest possü score, perhaps these should be used to anchor the scale. One final point about graphic displays of information must be made. Just as in writt The Denver Post circulation The Denver Post has shown steady ga ' '" --"*-—> the past two years Daily 260 250 240 230 ns in daily and Sunday circulation during Sunday 420 0 1989 1990 1991 1989 1990. SOURCE: ABC Publisher's Statements, six months ending September. 1991 450 The Denver Post Denver Post Circulation 2 200 3 150 100 50 4 19B9 1990 1991 ÍURE3.13 An Example of Exaggerating a Trend by Ignoring the Zero Point oF a Ratio Scale (top), Compared with a Fair Representation (bottom). Denver Post, November 3, 1991. 1'. 35 (A) Public School Funding Soars; No Payoff in SAT Scores c n 54,200------ ni = O-O *Q 54,000 S3.B0D $3,600 ■-53,400 BOO 1980 19B1 19BH 19B3 19S4 1985 19B6 19B7 19BB Year SAT Scores Soar Despite Minimal Gains in Funding (B) 520,000 r Si $17,500 3 ^■w S15,Q00 -•--5s S12.500 ao 3° 510,000 =5.1 S7.500 a. ~ S5.000 « 22,500 0 915 910 905 900 695 B90 V BB5 1980 1991 1982 1983 1984 19B5 1986 19B7 1 Year (C) Trends in Public School Fundings and SAT Scores 1600 1400 E 1200 oj 1000 BOO 600 400 19B0 1981 1982 19B3 19B4 1985 19BB 1987 1988 Year FIGURE 3.14 Illustrations of Propagandizing with Data (graphs A and B); Only Graph C Does not Distort the Information. 3.14 Case Study lililSPpM particularly wet or dry months. average, Mew York Ctty does not have any fH3 CHAPTER SUMMARY are grouped into inlervllIs anó disp|„ye[, „'|,icaU ' MCOm,!s w,denI ,rte observ.cons ^Z^^T^S^tS °! 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