The Estimation of Numerosity on Dot Maps

The Estimation of Numerosity on Dot Maps Christine L. Butler Special thanks to: Dr. Robert A. Rosing Mrs. Susan L. Parks Department of Geography and Regional Planning Salisbury State University Undergraduate Geography Major Department of Geography and Regional Planning Salisbury State College Salisbury, Maryland 21801 ABSTRACT An important concern for cartographers is in making maps that convey information accurately and efficiently. Because the dot map is a popular method of portraying qualitative data, it is necessary to know how effective the symbolization is in eliciting the desired intellectual responses. This paper investigates whether dot numerosity differences are over or underestimated in a purely map context. An experiment was designed to determine if dot numerosity differences are over or underestimated or if they are perceived accurately. The results of the test indicated that for an area with a given number of dots, comparisons made to areas with a higher number of dots are generally underestimated. Overestimation, on the other hand, was revealed when comparisons were made with areal units containing fewer dots. It has been suggested that numbers of dots be rescaled to decrease perceptual error on dot maps. These results support this suggestion. KEY WORDS : Dot Map, Numerousness, Apparent Density, Maps, Perception. PROBLEM SETTING It is the essential purpose of a map to produce, through visual means, desired intellectual responses on the part of the map user. Maps are effective communication devices only when they convey information efficiently and accurately. An important concern for cartographers is how the effectiveness of a map can be increased ; that is, how can a map be designed to result in more accurate and more efficient communication? A knowledge of how effective specific map symbols are in eliciting the desired intellectual responses is necessary. A variety of experiments have been implemented to assess the perceptual accuracy of some widely used map symbolization. One of the most wellknown is Flannery's work with graduated circles (1971). Flannery concluded that circle size differences were generally underestimated. He also made a plea for "continued and increased testing of 71

specific map symbols at all map user levels." The usefulness and popularity of the dot mapping method for portraying quantitative data by repetitive point symbols insures that many dot maps will be produced in the future. The dot map can show the details of a distribution of some phenomenon more clearly than any other type of map (Robinson 1984). It is understood by most readers because of its familiarity and the rather simple and direct relationship between the symbol and the mapped object (Provin 1977). The dot map is a well-established method for portraying many geographic distributions. Because they most commonly show distributions of discrete objects such as livestock or persons, agriculture is a frequent topic for dot maps (Fig. 1). An advantage of dot maps is the rel ative simplicity of design and ease of execution. One dot represents a specified number of objects and the total number of objects divided by this "value per dot," or unit value, gives the number of dots to be placed within any given subarea (Olson 1976). The optimum PEAS HARVESTED WISCONSIN..,... '.. '.::. '. '. '....... '. o 25 50 ONE DOT EQUALS FI FTY ACRES SOURCE - U.S. CENSliS OF AG., WIS.,1964 DB 72 Figure 1. An example of a dot map using agricultural data.

functioning of a dot map probably depends upon the reader's ability to react to two spontaneous stimuli : dot density, regardless of how large or small the region of that particular density; and, number of dots, regardless of how much area they occupy (Olson 1976). The term "number" refers to the actual count of individual objects in a collection. The estimation of the number of objects, without resorting to counting, is referred to by psychophysicists as numerousness or numerosity (Olson 1976). The number of objects should not be confused with the density of objects, the actual count of objects divided by the actual area. Apparent density would simply be the estimation of the number of objects divided by the estimated area. If there were 50 dots in a unit area and it is compared to 100 dots in a two-unit area, the 50 dots should elicit a "numerousness" response of "less" but an "apparent density" response of "same" (Olson 1976). These relationships are shown below (Fig. 2). Box A containing 50 dots has a lower number of dots than Box B, which contains 100 dots; however, both A and B have the same density. A 50 dots --- = 50 dots/unit 1 unit B 100 dots. --- = 50 dots/unit 2 units Therefore, Box A should elicit a "numerousness" response of "less than Box B," and Box A should elicit an "apparent density" response of "same as Box B." The early psychophysical literature indicates that the estimates of dot numerosity and/or density may be over or underestimated. Provin (1977) reviewed several experiments that showed varying results regarding tendencies to over or underestimate numerousness and/or density of dots. Messinger, as described by Provin, showed that if dots were scattered over a large area, their number is almost always overestimated and that increased scatter (more space between dots) tended to increase the amount of overestimation. In addition, increase in dot size resulted in greater overestimation (Provin 1977). An experiment by Mokre, also described by Provin, showed an opposite tendency. It was observed that, with larger dots and greater separation between the dots, numbers were perceived as smaller than when dots were smaller and less separated. Few studies dealing specifically with dot maps have been carried out by cartographers. Provin (1977) discussed an B A................. One Unit (Area) Two Units (Area) Figure 2. Boxes showing the differences between numerousness and apparent density. 73

experiment conducted by Bartz. She tested subjects' abilities to judge dot numbers and concluded that prediction of dot numbers is often underestimated. Olson tested subjects' abilities to judge dot density; however, the tests were not carried out in a mapping context but rather involved comparison of pairs of boxes containing a variable number of dots as Bartz has also done. Again, underestimation was apparent (Olson 1975). Sufficient evidence has been presented by psychophysicists to indicate that a relationship may exist between some of the design elements of a dot map and the perception of dot number. Provin (1977) tested individuals by using actual dot maps; however, blocks or boxes were put around the areas to be evaluated. Again, this test was not administered in an exclusively mapping context; that is, the blocks were added to the map which were not necessary in the relaying of the information of the map. STATEMENT OF THE PROBLEM The purpose of this study is to determine whether dot numerosity differences are over- or underestimated in a purely map context. The hypothesis is that when an area with a known number of dots is compared to an area with a larger number of dots this number will be underestimated. However, a similar comparison made to an area with fewer dots will result in an overestimation of numerousness. METHODOLOGY An experiment was designed by the author to determine if dot numerosity is over or underestimated or if it is perceived accurately. The subjects used in the experiments by Flannery (1971), Olson (1975), and Provin (1977) were students from different colleges and universities. This study followed the same procedure. The subjects were 73 students in introductory level geography courses at Salisbury State College in Salisbury, Maryland. The test instrument was a questionnaire (Appendix). It contained a brief description and an example of a dot map along with a general description of the exercise. The questionnaire contained two dot maps (Figs. C and D) of the United States which were produced using hypothetical data. Each map subject was given the number of dots in a particular state. Using that information, the student was asked to estimate the number of dots in five other states. Two of these states had a larger number of dots than the given state and two of them had a smaller number of dots. One of the states to be estimated had approximately the same number of dots as the given or standard state. In Part One of the questionnaire, the subjects tested were told that Nebraska contained 150 dots. They were asked to estimate the number of dots in South Dakota, Iowa, Kansas, Colorado and Wyoming. Colorado and South Dakota contained the largest number of dots (300 and 225), while Wyoming and Kansas contained the fewest number of dots (100 and 75). In Part Two of the questionnaire, subjects were told that Iowa contained 175 dots. The subjects were asked to estimate the number of dots in Wisconsin, Illinois, Missouri, Nebraska and South. Dakota. Nebraska (375 dots) and Illinois (250 dots) contained more dots than Iowa (175 dots). Also, South Dakota (150 dots) and Missouri (115 dots) contained fewer dots than Iowa. The questionnaire was pre-tested to check for any problems that may have existed in the instructions and understanding of the questions. No problems were found and the test was administered. RESULTS In Part One, the standard, Nebraska, contained 150 dots. When this state was compared with two other states, each containing more than 150 dots, the greatest percentages of numerosity estimates for these two states were below their actual values. For example, 54 percent of the estimates for Colorado were below its actual value of 300. The same was true for South Dakota which contained 225 dots; 60 percent of the estimates were below 225 (Fig. 3). 74

NUMEROSITY ESTIMATES STANDARD-150 30 "!20 225 dots '0 10 SOUTH DAKOTA IOWA 150 dots " 30 10 (; 100 50 PART ONE 350 4 0 250 300 350 400 KANSAS 75 dots COLORADO 300 dots E : 30 = 20 o 10 30 E20 _10 o o WYOMING 100 dots 30-10 o 100 150 X is the actual number of dots in each state o 50 12 0 200 Number of D ots Figure 3. Results from Part One of the test. When the standard of 150 dots was compared to Wyoming, which contained 100 dots, many of the estimates for Wyoming were accurate. This may be due to the fact that a "favorite" or " preferred number" was used for this state; that is, "100" for example, would likely be a more popular estimate than " 87." Less than one-third (26 percent) of the subjects estimated the number of dots in Wyoming as 100 while 40 percent of the estimates were higher than 100 and only 75

34 percent of the estimates were lower than 100 (Fig. 3). This shows that the estimates of numerosity for a state with fewer dots than the given state tend to be overestimated. Another example would be Kansas in Part One with 75 dots. The most common estimates for Kansas were 50, 75, and 100 because they are preferred numbers. Again, the greatest percent- NUMEROSITY ESTIMATES TANDARD-175" WISCONSIN E 30 " 20 175 dots '0 '0 PART TWO ~ ILLINOIS " 30 ;; E " 20 250 dots '0 '0 ~ '00 MISSOURI 30 ;; E 20 115 dots '0 10 ~ 100 " 30 NEBRASKA E 20 " 375 dots '0 10 ~ 50 200 " 30 SOUTH DAKOTA! 20 -;; 150 dots '0 10 ~ 100 X i s the actual number 50 250 of dots in each state Figure 4. Results from Part Two of the test. 76

age of estimates for Kansas, which has fewer dots than the given state, Nebraska, were above the actual number of dots. Nearly half (49 percent) of the estimates were above 75 while only 25 percent were below 75 (Fig. 3). The same results were apparent in Part Two. The standard in this case was Iowa with 175 dots. When a state with a larger number of dots was compared to Iowa, the number of dots in the first state was underestimated. However, a comparison of the standard with a state containing a fewer number of dots resulted in an overestimation of numerousness. The greatest percentage of estimates for Nebraska (375 dots) and Illinois (250 dots) were below their actual values. Twothirds (66 percent) of the estimates for Nebraska were below 375 and 40 percent of the estimates for Illinois were below 250. Likewise, the highest percentage of estimates for South Dakota (150 dots) and Missouri (115 dots) were higher than their actual values. More than half (58 percent) of the estimates for South Dakota were above 150 and 59 percent of the estimates for Missouri were above 115 (Fig. 4). CONCLUSIONS The findings of this study reveal that, for a given number of dots in one state, comparisons made to states with a higher number of dots than the given state are generally underestimated. They also show that comparisons of the given state to states containing fewer dots are overestimated. Olson (1976) has stated that: The fa ct that most people would estimate a field of 500 dots as less than 500 might be argued as irrelevant in that a dot map is a generalized portrayal that shows only more and less.. Although map readers will not absorb every detail on the map and will not be able to judge numbers and densities precisely, if a consistent bias is present in reader perception, it suggests that at least some improvement might be made (p. 126). Provin (1977) has made the suggestion that numbers of dots be rescaled to decrease perceptual error on dot maps. This study supports the suggestion that such rescaling is necessary. APPENDIX This exercise contains maps that use dots to show the location of some object in an area. This type of map is known as a dot map. Below is an example of a dot map (Fig. A) showing sheep and lambs on farms in 1964 where one dot represents 10,000 sheep and lambs. This map exercise is in two parts. You will be asked to estimate, in each part of this test, the number of dots in various states. The map of the United States (Fig. B) attached to this sheet is for your use during the testing to locate the states. You will be given 2'/2 minutes to complete each part of the test. Keep in mind that there are no wrong or right answers, so you should not feel pressured about getting a question wrong. Using the map above (Fig. C) and the map of the United States (Fig. B) attached to the instructions, complete the following: If NEBRASKA has 150 dots in its boundary, how many dots do the following states contain: 1. SOUTH DAKOTA? 2. IOWA? 3. KANSAS~?---------- 4. COLORADO? 5. WYOMING? Using the map above (Fig. D) and the map of the United States (Fig. B), complete the following: If IOWA has 175 dots in its boundary, how many dots do the following states contain: 1. WISCONSIN (WISC.)? 2. ILLINOIS (lll.)? 3. MISSOURI (MO.)? 4. NEBRASKA? 5. SOUTH DAKO-T-A-?------- STOP AT THIS POINT 77

. 10.000 he.ld U.s. TOTAL 15 41t.SOO 500 Figure A. An Example of a Dot Map. Use this map to lind the location 01 each state. Figure B. Locations of Selected States. 78

Figure C. Part One Map. Figure D. Part Two Map. 79

LIST OF REFERENCES Flannery, J. T., 1971. The Relative Effectiveness of Some Common Graduated Point Symbols in the Representation of Quantitative Data, The Canadian Cartographer 8 :96-109. Olson, J. M., 1975. Experience and the Improvement of Cartographic Communication, The Cartographic Journal 2 : 94-1 08. Olson, J. M., 1976. Rescaling Dot Maps for Pattern Enhancement, Internationales Jahrbuch fiir Kartographie 1: 125-137. Provin, R. W., 1977. The Perception of Numerousness on Dot Maps The American Cartographer 4 : 111-125. Robinson, A. H., Sale, R. D., Morrison, J. L. and Muehrcke, P. C., 1984. Elements of Cartography, 5th ed., John Wiley & Sons, Inc. New York. 80