All 4 books by Edward Tufte now in
paperback editions, $100 for all 4
Visual Display of Quantitative Information
Beautiful EvidencePaper/printing = original clothbound books.
Only available through ET's Graphics Press:
catalog + shopping cart
All 4 clothbound books, autographed by the author $150
catalog + shopping cart
Edward Tufte e-books
Immediate download to any computer:
Visual and Statistical Thinking $2
The Cognitive Style of Powerpoint $2
Seeing Around + Feynman Diagrams $2
Data Analysis for Politics and Policy $2catalog + shopping cart
Edward Tufte one-day course,
Presenting Data and Information
Tampa, March 1
Boston, March 14, 15, 16
Oakland, April 20
San Jose, April 21
Palo Alto, April 24
San Francisco, April 25, 26
My understanding is that rounding to two digits makes numbers easier to read, use and recall later. Certainly I find that to be the case. Obviously there are numbers that can't be rounded — bookkeepers have to account for every penny/cent, etc. But for general management purposes, marketing, decision-making, rounding can make data more lucid. The original reference I have on rounding to two digits for clarity is "Washburne, 1927". ( I don't know who Washburne was or what he/she actually proved or asserted.) Does anyone know of relevant research about rounding improving readability?
-- Sally Bigwood (email)
Two Google searches (the first for WASHBURNE 1927 and the second for WASHBURNE RANK ORDER) led me to this citation:
Washburne, J. N. (1927). An experimental study of various graphic, tabular and textual methods of presenting quantitative material. Journal of Educational Psychology, 18: 361-376, 465-476.
I'm unfamiliar with the article, but it may be the source of your original reference.
-- Mike Christenson (email)
See also A.S.C. Ehrenberg, "Rudiments of Numeracy," Journal of the Royal Statistical Society, A, 140 (1977), 277-297. On pages 281-282 there is a discussion of rounding to two significant digits. This reference is found on page 178 of Tufte's The Visual Display of Quantitative Information.
-- Bob Jones (email)
A small note: for those with access to JStor's online article database, the Ehrenberg article is available here.
-- Brooks Moses (email)
The number of significant digits depends on data underlying the calcuations. So for the magnetic moment of the election, the issue is what is going in something like the 20th significant digit. Most social sciences are probably one or two digit sciences, at least on those days that we get the sign right.
In the thread on cladograms, the article on the newly discovered euprimate skull says the skull is "54.97 Myr," which seems to be spuriously precise. Fossil science is probably good for one or two digits most of the time.
That is, the content should decide the number of significant digits. Rounding off for ease of reading is a different matter, perhaps important in journalistic accounts where irrelevant significant digits might well be truncated.
Finally we may want to reason quantitatively with one or two digit approximations. There are good examples of this kind of thinking in Feynman's books and in an article by Frederick Mosteller (in a book edited by Will Fairley that I can't find right now).
-- Edward Tufte
The two-significant-digit rule is more aptly described by Ehrenberg as the two-effective-digit rule or the two-variable-digits rule and consists of rounding data to two effective digits, that is digits which vary in that data. This amounts to having two digits in the residuals from the average.
2.3103, 2.3102, 2.3101, 2.3099, 2.3018, 2.3014, 2.2994, 2.2989. Average: 2.3052
Residuals (Deviation from the average)
-0.0051, -0.0050, -0.0049, -0.0047, 0.0034, 0.0038, 0.0058, 0.0063,
Therefore we need four decimal places to have two effective digits, both in the residuals and in the originals.
As E.T. has noted, the content determines significant digits. This is an accepted standard in the sciences (with slight variations on the details).
However, in the editorial arena, there is a general opinion that "presentation digits" should depend on the context, which would include not only the author's perspective, but the intended and anticipated audiences, intended and potential uses of the information, and the relative "value" of the lower digits in interpretation, among others.
If the data is presented merely as data, rather than information, without an intended purpose, e.g. to preserve it for potential future use, it might even be appropriate to preserve all measured digits, so that on future use the user could round and present according to the new context.
-- Bruce Hensley (email)
When I was an engineering student, one of my professors liked to say "if you care about the third significant digit of tensile strength, you are already in trouble."
-- MATTHEW D HEALY (email)
Accountants use degrees of rounding to distinguish between calculated figures and judgements.
"Our payroll bill for 2008 was ??137,384,272. In 2009, given the expected changes in the pattern of starters and leavers and other consequence of the new economic climate, it is expected to be ??0.12 billion.
-- Martin Ternouth (email)
When reporting data, it is well to remember that others may use the data later and may not be able to be in touch with you to ask questions. The usual rule of thumb is to report only as may digits as your measurement accuracy will support. For example, researchers are sometimes surprised to find that with sample sizes less than about 400, it is virtually meaningless to report more than one digit of a correlation coefficient, even though we all do it. (Remember that a quick and dirty way of estimating the standard error of a low correlation coefficient is to take the reciprocal of the square root of the sample size, so that with 400 cases the standard error is about .05.) This said, however, a colleague of mine in the 70s, the eminent nonparametric statistician James V. Bradley, pointed out that one reason in favor of reporting "extra" digits is that it is a way of minimizing the number of ties in data. Ties are (almost) always a problem in using distribution-free methods. BAF
-- Ben Fairbank (email)