Ranking Visualizations of Correlation using Weber’s Law Lane Harrison, Fumeng Yang, Steven Franconeri*, Remco Chang Tufts University, *Northwestern University 

Many techniques for the same data… 

Comparative Evaluation 

Comparative Evaluation … 8 C2 = 28 condition 1 condition 2 condition 3 conditions 

Model-based Evaluation 

Model-based Evaluation M1 Model the performance of scatterplots on a task. 

Model-based Evaluation M1 M2 M3 M4 M5 Model the performance of visualizations on a task. 

Model-based Evaluation M1 M2 M3 M4 M5 M3 M2 M4 M1 M5 > > > > 

Model-based Evaluation M1 M2 M3 M4 M5 M3 M2 M4 M1 M5 Models of performance can be compared directly and efficiently. Models are scalable and falsifiable. > > > > 

Model-based Evaluation M1 M2 M3 M4 M5 M3 M2 M4 M1 M5 To be effective, models must be grounded in theory. > > > > 

Model-based Evaluation M1 M2 M3 M4 M5 M3 M2 M4 M1 M5 To be effective, models must be grounded in theory. vs vs > > > > 

A model grounded in theory: The Perception of Correlation in Scatterplots Ron Rensink, Gideon Baldridge (2010) - Used psychophysiological methodologies. 

A model grounded in theory: The Perception of Correlation in Scatterplots Ron Rensink, Gideon Baldridge (2010) - Used psychophysiological methodologies. - Inferred just-noticeable differences for scatterplots depicting positive correlations. 

A model grounded in theory: The Perception of Correlation in Scatterplots Ron Rensink, Gideon Baldridge (2010) - Used psychophysiological methodologies. - Established that the perception of correlation in scatterplots can be modeled using Weber’s law. - Inferred just-noticeable differences for scatterplots depicting positive correlations. 

Psychophysiological methodology: 
 Which is more correlated? 

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Same difference
 but harder, why? 1.0 0.95 0.35 0.3 

just-noticeable differences (jnd) 

imagine yourself in a dark room… 

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JND: the smallest difference necessary to perceive two stimuli as being different. 

after many trials… 

0. 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND Rensink plotted JND as a function of correlation (r) (bright room) (dark room) better worse 

0. 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND To see a difference in data with correlation of 0.3, the comparison r must be +/- 0.2. (dark room) (bright room) better worse 

increase to 0.5! To see a diff, 

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decrease to 0.1! To see a diff, 

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Rensink’s insight… 

;ďͿƐĐĂƩĞƌƉůŽƚ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ;ĐͿƐĐĂƩĞƌƉůŽƚƌĞŐƌĞƐƐŝŽŶ rA 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ 0.0 0.00 0.06 0.12 0.18 0.24 Because the trend of JND & correlation is linear… 0 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND “The perception of correlation in scatterplots can be modeled using Weber’s Law.” (dark room) (bright room) better worse 

Weber's Law - sound, taste, weight, brightness, line-length - model for low-level perceptual discrimination ΔP = k * ΔI _ I 

Weber's Law - sound, taste, weight, brightness, line-length - model for low-level perceptual discrimination ΔP = k * ΔI _ I 

Weber's Law ΔP = k * ΔI I _ 

ΔP = k * ΔI I _ Perceived diff 

ΔP = k * ΔI I _ Perceived diff Actual intensity of Stimulus 

ΔP = k * ΔI I _ Actual intensity of Stimulus Change in Intensity Perceived diff 

ΔP = k * ΔI I _ Actual intensity of Stimulus Change in Intensity Perceived diff via experiment 

ΔP = k * ΔI I _ bright room, high intensity 

ΔP = k * ΔI I __ super bright light needed 

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Our hypothesis… 

;ďͿƐĐĂƩĞƌƉůŽƚ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ;ĐͿƐĐĂƩĞƌƉůŽƚƌĞŐƌĞƐƐŝŽŶ rA 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ 0.0 0.00 0.06 0.12 0.18 0.24 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND If the perception of correlation in scatterplots follows Weber’s law… better worse 

;ďͿƐĐĂƩĞƌƉůŽƚ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ;ĐͿƐĐĂƩĞƌƉůŽƚƌĞŐƌĞƐƐŝŽŶ rA 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ 0.0 0.00 0.06 0.12 0.18 0.24 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND What does the perception of correlation in other charts look like? better worse 

;ďͿƐĐĂƩĞƌƉůŽƚ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ;ĐͿƐĐĂƩĞƌƉůŽƚƌĞŐƌĞƐƐŝŽŶ rA 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ 0.0 0.00 0.06 0.12 0.18 0.24 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND vs What does the perception of correlation in other charts look like? better worse 

;ďͿƐĐĂƩĞƌƉůŽƚ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ;ĐͿƐĐĂƩĞƌƉůŽƚƌĞŐƌĞƐƐŝŽŶ rA 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ 0.0 0.00 0.06 0.12 0.18 0.24 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND What if the perception of correlation in other charts also follows Weber’s law? better worse 

;ďͿƐĐĂƩĞƌƉůŽƚ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ;ĐͿƐĐĂƩĞƌƉůŽƚƌĞŐƌĞƐƐŝŽŶ rA 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ 0.0 0.00 0.06 0.12 0.18 0.24 0.00 0.06 0.12 0.18 0.24 ;ĂͿƐĐĂƩĞƌƉůŽƚ͕ZĞŶƐŝŶŬ r 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 ƽ ƽ ƽ ƽ ƽ ƽ ƽ භ from above from below JND What if the perception of correlation in other charts also follows Weber’s law? vs better worse 

Experiment 

Stimuli: 
 9 chart types depicting 
 bi-variate data 

r = -1 r = -0.8 r = -0.3 r = 0.3 r = 0.8 r = 1 scatterplot parallel coordinates (pcp) stackedarea 

coordinates (pcp) stackedarea stackedline stackedbar r = -1 r = -0.8 r = -0.3 r = 0.3 r = 0.8 r = 1 scatterplot parallel coordinates (pcp) stackedarea 

radar line ordered line r = -1 r = -0.8 r = -0.3 r = 0.3 r = 0.8 r = 1 scatterplot parallel coordinates (pcp) stackedarea 

stackedbar donut radar line r = -1 r = -0.8 r = -0.3 r = 0.3 r = 0.8 r = 1 scatterplot parallel coordinates (pcp) stackedarea 

Methodology: Crowdsourced* on Amazon’s Mechanical Turk 

- n=1687 (AMT) - 9 charts - Normal data (in charts) - Between subjects - “Staircase” methodology - Kruskal-Wallis (overall) - Mann-Whitney 
 (post-hoc) - Bonferonni correction
 (p < 0.0036) Design Analyses 

Results 

scatterplot − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● scatterplot − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● parallel coordinates − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● parallel coordinates − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● stackedarea − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● stackedarea − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedline − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedline − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedbar − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedbar − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● donut − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● donut − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● line − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● line − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● radar − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● radar − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ordered line − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ordered line − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● 1. Inferred JNDs from experiment data for each chart and for 
 positive/negative correlations. 

scatterplot − positive r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● scatterplot − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● parallel coordinates − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● parallel coordinates − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● stackedarea − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● stackedarea − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedline − positive r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedline − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedbar − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedbar − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● donut − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● donut − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● line − positive 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● line − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● radar − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● radar − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ordered line − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ordered line − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● 

scatterplot − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● scatterplot − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● parallel coordinates − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● parallel coordinates − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● stackedarea − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● stackedarea − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedline − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedline − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedbar − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● stackedbar − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● donut − positive r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● donut − negative r jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● line − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● line − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● radar − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● radar − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ordered line − positive jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ordered line − negative jnd 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● 2. Tested Weber-model fit using previously- established methodologies. 1. Inferred JNDs from experiment data for each chart and for 
 positive/negative correlations. model fit results 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ordered line line radar stackedline stackedarea stackedbar donut scatterplot parallel coordinates scatterplot, Rensink positive negative JND rA 

model fit results 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ordered line line radar stackedline stackedarea stackedbar donut scatterplot parallel coordinates scatterplot, Rensink positive negative JND rA less precise more precise 

! !!! ! 

The perception of correlation 
 in every tested chart can be modeled using Weber’s law. 

You can do a lot of good with good models. 

1. Perceptually-backed ranking of precision for judging correlation 

model fit results 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ordered line line radar stackedline stackedarea stackedbar donut scatterplot parallel coordinates scatterplot, Rensink positive negative JND rA less precise more precise 

model fit results 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ordered line line radar stackedline stackedarea stackedbar donut scatterplot parallel coordinates scatterplot, Rensink positive negative JND rA Obtain rankings at each |r| 

Perceptually-backed ranking for vis of correlation 

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Interpolated (predicted) performance from models 

Overall ranking 

Perceptually-backed ranking for vis of correlation 

2. Explore properties of the perceptual space of correlation. 

r = -1 r = -0.8 r = -0.3 r = 0.3 r = 0.8 r = 1 scatterplot parallel coordinates (pcp) stackedarea 

0.7 0.8 0.85 scatterplot (positive) scatterplot (negative) parallel coordinates (negative) parallel coordinates (positive) Symmetric Not Symmetric ✓ ✓ ✓ X 0.7 0.8 0.85 ✓ Differences reliably 
 perceived Differences
 (not) reliably 
 perceived X 

0.7 0.8 0.85 scatterplot (positive) scatterplot (negative) parallel coordinates (negative) parallel coordinates (positive) Symmetric Asymmetric ✓ ✓ ✓ X 0.7 0.8 0.85 ✓ Differences reliably 
 perceived Differences
 (not) reliably 
 perceived X 

sp and pcp 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 parallelCoordinates scatterplot positive negative -PCP as good as scatterplots +PCP terrible 

To show correlations precisely in parallel coordinates plots, flip the axes to show as many negative correlations as possible. Design implication: scatterplot (positive) scatterplot (negative) parallel coordinates (negative) parallel coordinates (positive) scatterplot (positive) scatterplot (negative) parallel coordinates (negative) 

3. Guide the novice user in depicting correlation. 

stackedarea, stackedline and stackedbar 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 stackedline stackedarea stackedbar positive negative parallel coordinates (pcp) stackedarea stackedline stackedline stackedbar donut parallel coordinates (pcp) stackedarea stackedline stackedbar Worst Best OK 

Future Work 

Why does Weber’s law, which typically applies to low-level aspects of perception, work so well for modeling correlation? 

conj: People aren’t perceiving correlation, but the visual features related to correlation. 

r = 0.8 r = 0.5 

r = 0.8 r = 0.5 - - 

Theory-grounded models help build the science of visualization while providing actionable information to inform visualization design. To summarize: 

Data, Code, Analysis Scripts, Paper:
 github.com/TuftsVALT/ranking-correlation Lane Harrison ([email protected])