What scientific field(s) studies how people interpret quantitative summaries and visualizations? There's an abundance of well-known resources offering advice on data visualization. (E.g. Tufte, Stephen Few et al, Nathan Yau.) But to what field(s) might one turn to for answers to questions like these:


*

*Is the pie chart criticism relevant in practice? Are people that much better at interpreting linear scale length than arc length?

*Say I construct an index summary of a set of underlying variables, and explain to a lay audience that the United States has a 100 value in 2010, and a 110 in 2015. How will most people interpret these numbers? Are there natural cognitive habits that I should consider as I present this metric, either to leverage for better explanation or to guard against misinterpretation?


Put another way, to what scientific fields can presenters of quantitative information look for empirically sound and tested principles that help sort through the plethora of visualization and design advice available these days?
The aim is not to find advice, ideas or current consensus on how best to visualize data or approach novel data visualization problems, but to learn where to look for the science of how people interpret quantitative and/or visual information.
(Extra credit for references to journals, conferences and scholars of the field.)
 A: Gerd Gigerenzer is widely acknowledged as one of the world experts in the cognitive aspects of numeracy or, alternatively, innumeracy. He has many papers and books on these topics referenced on his website (https://www.mpib-berlin.mpg.de/en/staff/gerd-gigerenzer). One of his key texts is his 2002 book Calculated risks: How to know when numbers deceive you. Read the abstract here: https://www.mpib-berlin.mpg.de/en/research/adaptive-behavior-and-cognition/publications/books/calculated-risks
Related to Gigerenzer's work is cognition-based decision theoretic work that looks at the way information is presented. A representative paper here is Dan Goldstein's The Illusion of Wealth and its Reversal available here ... http://rady.ucsd.edu/docs/seminars/goldstein.pdf Here's from the intro:

Recently, researchers and policy makers have started to pay more
  attention not just to choice architecture but also to information
  architecture: the format in which information is presented to people.
  Research in information architecture has shown, for example, that the
  caloric content of food can be well appreciated in terms of the amount
  of exercise it would take to work calories off, and the comprehension
  of cars’ energy efficiency can be enhanced by presenting information
  in terms of gallons per 100 miles instead of miles per gallon. This
  paper investigates information architecture, though instead of
  consuming calories or gasoline, we address economic consumption in
  retirement.

An important recent addition to the literature is Berkeley Dietvorst's research into "algorithm aversion" and decision-making. Dietvorst contends that wrt predictive modeling, the technically naive and/or illiterate tend to assume that predictive models are a "magic bullet" or perfectly informative and when the algorithms prove to be, at best, weakly predictive, then the typical response is to reject quantitative solutions altogether.
https://marketing.wharton.upenn.edu/mktg/assets/File/Dietvorst%20Simmons%20&%20Massey%202014.pdf
Then there are bloggers like Kaiser Fung who maintains his Junkcharts website critiquing the graphs and visualizations of major pubs such as the NYTs or the WSJ
http://junkcharts.typepad.com/ 
Related to your question of visualization is the work of design experts such as Manuel Lima who maintains a website VisualComplexity.com covering the many approaches to this. Lima also teaches data visualization at Parsons School of Design in NYC. 
http://www.visualcomplexity.com/vc/
Besides Parsons, other design and visualization institutions include:
College of Design and Social Context
https://www.rmit.edu.au/about/our-education/academic-colleges/college-of-design-and-social-context/ 
UCLA's Culture Analytics Institute
http://www.ipam.ucla.edu/programs/long-programs/culture-analytics/
Google's Cultural Institute
https://www.google.com/culturalinstitute/home
A MoMA design exhibition and book
http://www.moma.org/calendar/exhibitions/1071?locale=en
http://www.amazon.com/Talk-Me-Communication-between-Objects/dp/0870707965
In terms of conferences there is the Eyeo Festival
http://eyeofestival.com/
In R software, the visualization guru is Hadley Wickham
http://had.co.nz/
In SAS software, there is Rob Allison
http://www.robslink.com/SAS/graph_book.htm
Finally, there are no shortage of "one-off" kinds of websites:
http://infosthetics.com/  great visuals of govt data
http://www.thefunctionalart.com/2012/09/in-praise-of-connected-scatter-plots.html
http://www.informationisbeautifulawards.com/
How to display data badly by Karl Broman
https://www.biostat.wisc.edu/~kbroman/presentations/IowaState2013/graphs_combined.pdf
https://www.biostat.wisc.edu/~kbroman/presentations/IowaState2013/index.html
Maria Popova's Design and Communication blog
https://www.brainpickings.org/2012/06/26/talk-to-me-moma-paola-antonelli-book/
Gallery of Data Visualization
http://www.datavis.ca/gallery/index.php
Periodic Table of Data Visualization
http://www.visual-literacy.org/periodic_table/periodic_table.html
Our World in Data
http://ourworldindata.org/
This just begins to scratch the surface of what's out there...
A: Psychophysics studies how humans respond to and interpret stimuli, to include interpretation of data visualizations. The Cleveland and McGill paper linked in the comments is an example, and the second section of this paper gives a quick overview of a few perspectives.
Numerical or mathematical cognition is a sub-discipline of cognitive science that studies things like number sense. It sometimes borrows concepts from psychophysics, for instance Fechner's scale, which "states that subjective sensation is proportional to the logarithm of the stimulus intensity." Wiki's description of the concept applied to numerical cognition:

Psychological studies show that it becomes increasingly difficult to discriminate among two numbers as the difference between them decreases. This is called the distance effect. This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances. It may also play a role in explaining why consumers neglect to shop around to save a small percentage on a large purchase, but will shop around to save a large percentage on a small purchase which represents a much smaller absolute dollar amount.

Related, in behavioral economics, prospect theory (original paper) examines human choices between risky, probabilistic alternatives. 
