This document summarizes a psychophysical study that investigated how accurately people can estimate the relative sizes of elementary physical shapes. The study presented participants with pairs of bars and spheres of varying sizes and asked them to estimate size ratios using two methods. For bars, estimates were highly accurate and linear regardless of encoding method. For spheres, encoding by volume led to misleading underestimates, while surface area encoding produced more accurate overestimates. The study provides insights into how different physical variables influence size perception and recommendations for effectively encoding data physically.

1. A Psychophysical Investigation of Size
as a Physical Variable
Yvonne Jansen
Kasper Hornbæk

2. Data Physicalizations
"a physical artifact whose geometry
or material properties encode data."
Atelier Brückner, German Age Pyramid, 2013

3. Data Communication
"a physical artifact whose geometry
or material properties encode data."
Data source: dataphys.org
Data Physicalization

4. Data Communication
Hans Rosling, gapminder.com
Presentations

5. Data Communication Presentations Exhibitions
Joyce Hsiang & Bimal Mendis, The City of 7 Billion, 2011-2013
Atelier Brückner, German Age Pyramid, 2013

6. Loren Madsen, Chicago crime rates, District 5 police station, 2014
Presentations Exhibitions Public Installation

7. Data
Physicalizations
Physical VariablesVisual Variables
Data
Visualizations

8. How should we encode data?

9. Data Visualization
Tableau's Show Me feature
[Source: onlinehelp.tableau.com]
1980s ranking of visual variables!
! ! (Cleveland & McGill 1982-1987,
Mackinlay 1986, Spence 1990)
1926 circle vs bars debate
(e.g., Eels 1926, Croxton & Stryker 1927,
Croxton & Stein 1932)
bibliography under yvonnejansen.me/size

10. Data Physicalization
Image source:
Kahrimanovic et al, 2010
Psychology & Psychophysics experiments
1970 Baird, Psychophysical Analysis of Visual Space
2010 Kahrimanovic et al., haptic volume perception
1969 Stanek, volume and surface judgments
e.g.,

11. sequential presentation
size 10 size ? size ?
…

12. Source:
balanceandmobility.com
Source:mv.cvc.uab.es
Chin rests
(see Baird 1970)
Free head movements

13. Instructions matter
how large is the smaller shape (compared to the larger)?
how large appears the smaller shape?
[Teeghtsoonian, The Judgment of Size, 1965]

14. ratio between heights
How should we describe size?
11 cm 3 cm

15. ratio between ?
How should we describe size?

16. ratio between
(in the store) diameters
6 cm 2 cm
33 %
How should we describe size?

17. (in the store) diameters
113 cm3 4.2 cm3
33 %
3.7 %
How should we describe size?
ratio between
("commonly") volumes

18. (in the store) diameters
("commonly") volumes
(also possible) surface areas
113 cm2 12.6 cm2
33 %
3.7 %
11.1 %
How should we describe size?
ratio between

19. How should we encode data
physically?
Joyce Hsiang & Bimal Mendis, The City of 7 Billion, 2011-2013

20. Research Questions
1. How accurately are elementary shapes estimated?
3. Are estimates systematically biased?
2. How similar are estimates between individuals?

21. Experiment

22. Bars Spheres

23. Bars Spheres
see dataphys.org/list

24. 28 pairs
7 sizes diameters 1.2-7cmheights 1-15cm
10 participants

26. Estimation Methods
Ratio estimation
e.g., Cleveland & McGill (1984)
Constant sum
e.g., Spence (1990)

27. 2 methods
Constant sum
e.g., Spence (1990)
represents larger shape
shorter shape
Ratio estimation
e.g., Cleveland & McGill (1984)

28. Task
"Indicate the percentage of the quantity represented by the
smaller shape relative to the larger shape."

29. "Divide the line such that the left part represents the quantity represented by the
left shape and the right part represents the quantity of the right shape.”
Task
quantity represented
by larger shape
…shorter shape

30. Results

31. bar sphere
0
25
50
75
100
0
25
50
75
100
constantsumratioestimation
0 25 50 75 100 0 25 50 75 100
true ratio
estimatedratio
method
shape
estimatedratio
true ratio

32. bar sphere
0
25
50
75
100
0
25
50
75
100
constantsumratioestimation
0 25 50 75 100 0 25 50 75 100
true ratio
estimatedratio
method
shape
estimatedratio
true ratio
y = xa
(Stevens’ law)

33. bar sphere
0
25
50
75
100
0
25
50
75
100
constantsumratioestimation
0 25 50 75 100 0 25 50 75 100
true ratio
estimatedratio
method
shape
estimatedratio
true ratio
y = xa
(logistic curve)
(Stevens’ law)
y =
1
10
b+a·log 1
x −1
+ 1

34. bar sphere
0
25
50
75
100
0
25
50
75
100
constantsumratioestimation
0 25 50 75 100 0 25 50 75 100
true ratio
estimatedratio
method
shape
estimatedratio
true ratio
y = xa
y =
1
10
b+a·log 1
x −1
+ 1
(logistic curve)
(Stevens’ law)

35. Accuracy

36. Accuracy
bars: ratio between heights
spheres: ratio between volumes, diameters, and surface areas

37. absolute discrepancy (in percent)
Accuracy
2D bars (Spence)
3D bars (height)
diameter1.6
surface
diameter
volume
0 5 10 15 20
(error bars indicate 95% bootstrapped confidence intervals)

38. Bias in Estimates
residuals

39. Bias in Estimates
−50
−25
0
25
50
0 25 50 75 100
ratio between volumes (in %)
residuals
volume-based
encoding
Spheres

40. Bias in Estimates
−50
−25
0
25
50
0 25 50 75 100
ratio between volumes (in %)
residuals
−50
−25
0
25
50
0 25 50 75 100
ratio between diameters (in %)
residuals
volume-based
encoding
diameter-based
encoding

41. −50
−25
0
25
50
0 25 50 75 100
ratio between surfaces areas (in %)
residuals
Bias in Estimates
−50
−25
0
25
50
0 25 50 75 100
ratio between volumes (in %)
residuals
volume-based
encoding
surface-based
encoding

42. estimates ~ linear
Take Aways – Bars
response curves similar across people

43. Take Aways – Bars
●
●
●
●
●
●
line (H) length
line (V) length
bar height
box height
cylinder height
bar (RE) height
0 2 4 6
Spence(1990)
lower accuracy than 2D bars
but still < 5%

44. Take away - Spheres
Encoding in volumes misleading representations✘
diameter only marginally better
surface area
✘
much better (but still more overestimations)

45. Take away - Spheres
Encoding in volumes misleading representations✘
diameter only marginally better
surface area
✘
much better (but still more overestimations)
supported by haptic perception studies
(Kahrimanovic et al, 2010)
see bibliography under yvonnejansen.me/size

46. Open Questions / Future Work
Other physical shapes
Interactions between vision and touch
Large range of absolute sizes

47. Raw data, R scripts, and additional charts at
yvonnejansen.me/size
(work in progress)
yvja@di.ku.dk
Comments, questions, requests