the visual display of quantitative information...visual display of quantitative information" by...

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The Visual Display of Quantitative Information Marvin A. Ruder Heidelberg University Winter term 2019/20 Seminar “How do I lie with statistics?”

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Page 1: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

The Visual Display of Quantitative Information

Marvin A. Ruder

Heidelberg UniversityWinter term 2019/20

Seminar “How do I lie with statistics?”

Page 2: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

Abstract

In this report, different types of graphical integrity are presented. We showhow newspapers and magazines use different techniques to create graphicsthat lack graphical integrity and therefore influence the readers’ opinion on amatter. We present methods that help creating integer graphics and extractuseful information from non-integer graphics.

This report is part of the winter term 2019/20 seminar “How do I liewith statistics?” at Heidelberg University. It is based on the book “TheVisual Display of Quantitative Information” by Edward R. Tufte [1].

Page 3: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

Chapter 1

Graphical Integrity

When looking up graphical integrity in adictionary, one will find a definition such as “thequality of being honest and having strong moralprinciples”. It follows that a graphic that lacksintegrity is in some way dishonest and immoral,a technique often used by graphic designersto influence the opinion of the audience of agraphic on a certain topic, this opinion beingdifferent than the opinion the audience woulddevelop from the data the graphic is basedon. In this section, it is discussed how thelack of graphical integrity can be detected, andmethods on how to create integer graphics arepresented.

1.1 Distortion in Graphics

In general, a graphic can be considered as dis-torting whenever its visual representation isnot consistent with the numerical representa-tion of the data it visualizes. While the nu-merical representation can be obtained easilyand unambiguously, the term “visual represen-tation” can be interpreted in different ways.

Let us assume an example graphic that vi-sualizes data using the area of two-dimensionalshapes, as can be found in Figure 1.1.

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Figure 1.1: A graphic using rectangular areasfor data visualization.

A simple understanding of visual represen-tation would consider the actual area of theshape that everyone can measure using a ruleras representative for the visualized data.

A different approach would consider theperceived visual effect of the area difference, asshown in Figure 1.1. Here, we assume that peo-ple will systematically over- or underestimatea relation in area size.

To decide which concept of visual represen-tation helps at maintaining graphical integrity,we need to analyze the perceived visual effect.For this, we will look at some experimentalresults. During an experiment, a large numberof people were given circles of different sizesand were asked to guess the area of the circles.From the answers, the following power law hasbeen derived:

reported perceived area = (actual area)x,

x = 0.8 ± 0.3 (1.1)

From the exponent x being between 0 and 1we can see that the perceived area grows moreslowly than the actual area, and the average per-son underestimates relations in area size. Also,from the comparably large uncertainty factorof 0.3, we can deduct that different people havevery different perceptions of area sizes. Whilesome people made no difference between the re-lation of the circle areas and the relation of theone-dimensional circle diameters (happening atx = 0.5), some people even overestimated therelation of the circle areas (at x > 1).

The only solution to this dilemma is to notuse the perceived visual effect of an area asvisual representation in graphics. From this,[1] deducts the following principles that shallenhance graphical integrity:

1

Page 4: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

“The representation of numbers,as physically measured on the sur-face of the graphic itself, should bedirectly proportional to the numer-ical quantities represented.

Clear, detailed, and thoroughlabeling should be used to defeatgraphical distortion and ambiguity.Write out explanations of the dataon the graphic itself. Label impor-tant events in the data.” [1]

1.1.1 The Lie Factor

When analyzing the integrity of graphics, a mea-surement of the distortion in graphics would behelpful. For this, the Lie Factor is introduced.It can be computed using the equation

Lie Factor =size of effect shown in graphic

size of effect in data.

(1.2)It is desirable that both effects are of the

same size, following the previous principle. Alie factor of 1 thereby certifies that the designerdid a good job and did not use distortion in hisgraphic. However, if the lie factor is larger orsmaller than 1, we can infer that the graphicdistorts by overstating or understating the visu-alized effect in the data respectively. Generallyspeaking, most distorting graphics involve over-stating.

The following example uses the lie factor todescribe the distortion involved. In Figure 1.2,the fuel economy standards (in miles per gal-lon1) between 1978 and 1985 set for cars by the

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1While the unit miles per gallon will confuse mostreaders in Germany—as the common unit for fuel econ-omy in Germany is liters per (typically 100) kilome-

U.S. Congress and the Department of Trans-portation are visualized using the lengths ofhorizontal lines along a street. While 18 mpgin 1978 are represented using a 0.6 in long line,27.5 mpg in 1985 are visualized using a line5.3 in long. From the numbers alone, we canimmediately see that this graphic distorts theunderlying numbers.

Using Equation 1.2, we can also quantifythis distortion. From 1978 to 1985 the fuel econ-omy standard increases by 53 percent—from18 mpg to 27.5 mpg. The lengths of the cor-responding numbers however increase by 783percent, from 0.6 in to 5.3 in. The two ratioscan be used to calculate the lie factor as

783%

53%= 14.8. (1.3)

When we take the information from Fig-ure 1.2 to create an integer graphic, shownin Figure 1.3, more information is instantly re-vealed, such as periods of slow or rapid increasesof the fuel economy standard. Adding contex-tual information such as the fuel efficiency ofthe cars currently in use can help the audiencegetting a better understanding of the depictedinformation.

1.2 Data and DesignVariation

When humans look at a graphic, they haveexpectations for its consistency. For exam-ple, patterns that extend over some parts of

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Figure 1.3: An integer graphic showing theinformation from Figure 1.2.

ters— the average New York Times reader will notbe distracted by this unit, so using this unit does notconflict with graphical integrity.

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Figure 1.4: National Science Foundation,Science Indicators, 1974 (Washington, D.C.,1976), p. 15.

a graphic are expected to continue along thewhole graphic. Following that, violating thispattern will create wrong expectations andleads to deception.

Let us take a look at the example in Fig-ure 1.4. It visualizes the nobel prizes won byscientists from a number of countries in thetime range from 1901 to 1974.

From the rapid decline in the last time in-terval, one might assume that much less nobelprizes were awarded to the scientists of theevaluated contries during the 70s. As somepeople may now speculate about the reason forthis decline and may conclude that all scien-tists suddenly became dumb or died during agigantic scientific accident, the keen and pre-cise observer will discover that the last timeinterval contains only four years between 1971and 1974, whereas all other time intervals areten years wide.

In this example, the consistent width of thetime interval is a pattern, which is interruptedwith the last time interval. However, the au-dience expects this last interval to be also tenyears long, which creates deception.

Here, we can split the parts of the visual-ization into two groups: data variation anddesign variation. Data variation refers tothe different values displayed in the chart vary-ing over time, which is what the audience isinterested in. On the other hand, design varia-tion describes the change in the time intervalsize. As demonstrated here, design variation isan unwanted technique and should be avoided,which leads to the simple principle

“Show data variation, not designvariation.” [1]

When design variation is removed from theprevious example using time intervals of equalsize, the visualization appears far more reason-able, as can be seen in Figure 1.5

Another example shows design variation atan alarming rate. In Figure 1.6, the OPEC oilprice is visualized using a bar chart. Immedi-ately visible are the two different parts of thegraphic, where bars represent either yearly orquarterly oil prices. In addition to that andwithout any special notice, the vertical scaleof the four quarterly bars is different for everyindividual bar. The different time and pricescales are shown in Table 1.1.

With this variety of scales, it is impossibleto compare one bar from 1979 against any otherbar in the chart, which makes the graphic quiteuseless.

In another example, we take a look at de-sign variation in three dimensions. Figure 1.7

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3

Page 6: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

During this time one vertical inch equals1973–1978 $8.00Jan–Mar 1979 $4.73Apr–Jun 1979 $4.37Jul–Aug 1979 $4.16Oct–Dec 1979 $3.92

During this time one horizontal inch equals1973–1978 3.8 years1979 0.57 years

Table 1.1: An overview on the different scalesin Figure 1.6.

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visualizes the same oil prices as discussed beforeusing oil barrels. While the barrel representingthe first year appears to be far away from theviewer, the most recent barrel is depicted veryclose. The usage of perspective is a kind ofdesign variation, resulting in a lie factor of 9.4.

In addition to the examples shown, somenewspapers and magazines created good andinteger graphics showing the oil price in the1970s, as in Figure 1.8. We can see that notonly the nominal oil price is given, but alsothe real price adjusted for inflation. From Fig-ure 1.8a, we can see that the real oil price wasdecreasing from 1974 to 1978, while Figure 1.6shows increasing bars during that time, makingit impossible for the audience to recognize thisfact. The example of inf lation points out thatin some cases, we are interested in removingnot only design variation, but also some waysof data variation, from our graphics in order tosustain graphical integrity.

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Figure 1.8: Good visualizations of oil prices.

4

Page 7: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

1.3 The Case ofSkyrocketingGovernment Spending

Before we will discuss the effect of inflation inthe following example graphic from Figure 1.9,we will take a look at some graphical gimmicksthat secretly let the values look more dramati-cally than they actually are.

In the example, we see the increasing totalbudget expenditures of the State of New Yorkfrom the fiscals 1966 to 1976. In the bottomright corner, two arrows are pointing upwardsat the corresponding bars to indicate that theycontain “estimated” and “recommended” val-ues. Although those words could have easilybeen used without the arrows, the chart de-signer leads the viewers’ attention towards thelargest two bars in the graphic and therebycreates a certain impression as if the latest ex-penditures were larger than they actually areand weightier than those before.

To find the next grapical gimmick, we needto pay attention to the three-dimensional designof the graphic. When looking closely at the topof the fiscal 1974’s bar, it appears that this bar,together with the bars to its right, is in frontof the bars left to itself. Again, this creates animpression as if the corresponding expenditures,being the highest in the graphic, were somewhatmore important.

The last design element distracting the view-ers’ attention and concentration is the three-dimensional design itself, which creates a veryfuzzy and restless graphic. When removing theperspective elements as in Figure 1.10, a muchcalmer chart emerges.

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All of the aforementioned graphical tech-niques lead the viewers’ attention away fromthe underlying numbers and create a drama-tizing effect, overstating the “skyrocketing” in-crease in government spendings.

Apart from the graphical distractions, twounwanted effects of data variation are part ofthe chart. The first one, as introduced before,is inflation. 1 US Dollar in 1966 has about thesame worth as 2.03 US Dollar in 1977.

Another effect to consider is the growth ofthe population, as more people using a state’sinfrastructure or public services justify largergovernment expenditures. In the State of NewYork, the population increased by about 10percent in the corresponding years.

We can take both effects into account by vi-sualizing the per capita budget expendituresin constant dollars, as shown in Figure 1.11.It can be seen that, apart from a 20 percentincrease by 1970, the spendings were almostconstant, remaining within a 5 percent intervaluntil 1977; a result fundamentally different tothe first impression from Figure 1.9.

Therefore, to enhance graphical integrity,the following principle is to be obeyed:

“In time-series displays of money,deflated and standardized units ofmonetary measurement are nearlyalways better than nominal units.”[1]

Figure 1.10: New York Times, February 1, 1976,p. iv-6, edited by Tufte.

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Figure 1.11: An integer graphic showing theinformation from Figure 1.9.

5

Page 8: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

1.4 Visual Area andNumerical Measure

As already discussed in section 1.1, several prob-lems can occur when numbers are visualizedusing areas of figures. This holds particu-larly true when the visualization is done wrong.The core problem is that in most cases, one-dimensional data is presented using two or moredimensions. When doing so, the area needs tobe proportional to the data it represents, asconcluded in section 1.1. Very often though,the width and height of an area are scaledlinearly with the data, so the area increasesquadratically, leading to misrepresented data.

When looking at Figure 1.12, we see thatthe size of the doctor is supposed to be consis-tent with the “percentage of doctors devotedsolely to family practice”. When we comparethe numbers from 1990 and 1964, we can saythat the doctor on the left should be 2.25 timeslarger than the right hand side doctor. How-ever, it can be seen with the naked eye thatthe difference in height of the doctors alreadyroughly corresponds to the factor of 2.25, sothe difference in area is close to a factor of 5.As the human eye pays attention to the areaof a depicted object rather than to its height,the 1964 doctor also appears to be far largerin comparison to his 1990 colleague than heshould be. In our example, a lie factor of 2.8results from the falsely scaled doctors.

The same mistake that has been done in theprevious example is even more serious when it isdone in three dimensions. Let us take a look at

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the example from Figure 1.7 again. Its lie factorof 9.4 that has been calculated in section 1.2is based on the assumption that the viewer issupposed to interpret the printed area of anoil barrel as a measure for the underlying data.But if we take the three-dimensional depictionseriously and assume that the volume of a barrelshall correspond to the data, an astoundingincrease of 27000 percent can be calculatedbetween the smallest and the largest barrel,whereas the underlying numbers increase byonly 454 percent. Following that, a lie factor of59.4 can be calculated, which must be a record.

As we have already seen in section 1.1, largedifferences occur in how people perceive thesize of areas and different impressions may arisefrom the same graphic. Following that, showingone-dimensional information using two or evenmore dimensions is an inefficient technique witha large perception bias which is also often donewrong. To enhance graphical integrity here, weneed to follow the principle

“The number of information-carrying(variable) dimensions depicted shouldnot exceed the number of dimen-sions in the data.” [1]

However, not all two-dimensional represen-tations of data are misleading. With Figure 1.13,

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Figure 1.13: Antonio Gabaglio, Teoria Gen-erale della Statistica (Milan, second edition,1888).

6

Page 9: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

we have a good example of using two dimen-sions for data visualization. The rectanglesshow both the number and average depositvalue of postal savings books in Italy from dif-ferent months in the late 19th century. Thearea of the rectangles thereby show the totaldeposit of all postal savings books. In this ex-ample, two dimensions are used to visualizetwo-dimensional data, in accordance with theprinciple above.

1.5 Context is Essentialfor Graphical Integrity

Quite often, graphics which seek to influencethe opinion of the audience do not show falseinformation, but omit certain information suchthat only information that the chart designerlikes is presented.

Figure 1.14 shows the traffic deaths in thestate of Conneticut at two points in time, 1955and 1956, and provides the information thatin 1956 the police started to prosecute driversexceeding speed limits more strictly. The datapoint from 1955 is larger than the point from1956, showing a decrease in traffic deaths. Atfirst sight, one might conclude that the stricterpolice enforcement is responsible for the trafficdeath decline and therefore was quite successful.But without any knowledge about traffic deathsin other years, one cannot be certain that thedecrease is significant and not similar to otherchanges in the years before or after.

To help us interpret the context of certaininformation, Figure 1.15 provides us with threepossible context data sets for our two datapoints. Very different interpretations arise fromthe different scenarios.

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In the first scenario, a increase or decreaseby the same amount as in Figure 1.14 occursevery year and the change would not be signifi-cant, so the police enforcement would not havecaused the decline in traffic deaths.

The next scenario shows a peak—the trafficdeath toll increases in one year and decreasesby the same amount in the following year. Apossible reason for this could be a one-timeevent such as extreme weather conditions ora mass collision. Following that assumption,the numbers would go back to normal in thefollowing year by default, with or without thestronger enforcement against speeding. Again,the police enforcement would not have causedthe decline in traffic deaths.

However, the last scenario shows a constanthigh number of traffic deaths in the years upto 1955 and a constant low number of trafficdeaths in the years after 1956. Here, the policeenforcement might have caused the decline intraffic deaths, however, we have insufficientinformation on the matter to say that for sure.

Figure 1.16 shows the information from Fig-ure 1.14 together with the traffic deaths from1951 to 1959. From there, we can see thatthe decrease is of a similar magnitude to otheryears’. Also, the 1955 traffic deaths are thehighest in this decade, so the scenario is similarto a peak as described before.

When we add even more data from the sametime, but from different states, we see in Fig-ure 1.17 that not only did the 1956 numbersdecline in Conneticut, but also in three otherstates. It is therefore very likely that the de-clining traffic deaths have nothing to do withthe stronger police enforcement in Conneticut,

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Page 10: The Visual Display of Quantitative Information...Visual Display of Quantitative Information" by Edward R. Tufte [1]. Chapter 1 Graphical Integrity When looking up graphical integrity

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Figure 1.17: Donald T. Campbell and H.Laurence Ross, “The Connecticut Crackdownon Speeding: Time Series Data in Quasi-Experimental Analysis,” in Edward R. Tufte,ed., The Quantitative Analysis of Social Prob-lems (Reading, Mass., 1970), 110-125.

but are provoked by some cause that affectsmore than one state.

Here, using the contextual data points inFigure 1.17, we were able to conclude whatwould not be possible from the two data pointswe started with. Therefore, when graphicalintegrity shall be preserved, we need to followthe principle that

“Graphics must not quote data outof context.” [1]

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Chapter 2

Sources of Graphical Integrity

In the previous chapter we have seen thateven respected and well-known newspapers ormagazines print graphics that lack graphicalintegrity and show a distorted picture of reality.So why do artists draw graphics that lie, andwhy do the world’s major newspapers and mag-azines publish them? Do they really want tomanipulate their readers picture of the world,do they use misleading graphics unintention-ally, or are they just not interested in integergraphics?

In [1], three explanatory approaches arepresented to answer the questions above.

The Lack of Quantitative Skills of Pro-fessional Artists Most designers that cre-ate charts and graphics for newspapers studiedfine arts, so they are skilled to let graphics lookappealing and interesting. However, artists usu-ally have no experience with analyzing data andsimply do not know how to achieve graphicalintegrity. Their goal is to create fancy graphics,not accurate graphics.

The Doctrine That Statistical Data AreBoring From this doctrine, it follows thatgraphics are used for the purpose of catchingthe reader’s attention and being entertaining,rather than informing. A chart specialist ofthe Time magazine, who also was an art-schoolgraduate, was quoted: “The challenge is topresent statistics as a visual idea rather thana parade of numbers.” From here, we can seethat the designer had no intentions in graphicalintegrity, a consequence of letting designerscreate graphics and not statisticians, who knowhow to work with numbers and present themin an accurate way, or authors, who know thematter of the article the graphic belongs to andcan put the numbers in a context.

The Doctrine That Graphics Are Onlyfor the Unsophisticated Reader Some ed-itors use graphics to entertain those peoplefrom an audience from whom it is assumedthat they will not understand the informationor the words in the articles. While the in-telligent readers shall read the text, the lessintelligent readers are supposed to get a briefidea of the matter from the graphic that iscreated in a way that makes it simple to un-derstand the information presented. For thisreason, overstating effects is a commonly usedtechnique in order to emphasize certain circum-stances. A publisher of a magazine intendedfor children, who are a good example for lesssophisticated readers, once said that they “pro-duced an article that was longer on graphicsthan on information. We had feared childrenmight be overwhelmed by too many facts.” Be-cause of this fear, the publisher did not makeany effort to create integer graphics, since thechildren will see no difference between thoseand non-integer graphics.

From the approaches above, [1] comes tothe conclusion that

“Graphical competence demands threequite different skills: the substan-tive, statistical, and artistic. Yetnow most graphical work, particu-larly at news publications, is un-der the direction of but a singleexpertise—the artistic. Substan-tive and quantitative expertise mustalso participate in the design of datagraphics, at least if statistical in-tegrity and graphical sophisticationare to be achieved.” [1]

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Bibliography

[1] Edward Rolf Tufte. The Visual Display of Quantitative Information. 2nd ed. Cheshire,Conneticut, U.S.A.: Graphics Press, 2001. Chap. 2–3, pp. 53–87. 197 pp. isbn: 978-0-9613921-4-7.

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