# Should percentages be reported with decimal places?

When presenting data using a percentage, is it a good thing to have decimal places, say 2 decimal places instead of rounding off to whole numbers?

For example, instead of 23.43%, you round off to 23%.

I am looking at this from the perspective of whether the 2 decimal places accuracy will make much difference since we are dealing with percentage and not raw data value.

• There are many fields where small percents are so small that people use parts per million, per billion and so forth. Either that's a different question -- because people do or should know that citing numbers like 0.000001 or even 0.0001% is in that circumstance silly and one should use different units -- or it's another answer to this question. When some or all of the numbers of interest are very small, large numbers of decimal places may be essential as well as informative. Nov 8 '18 at 17:47
• This is a special case of the issues discussed at stats.stackexchange.com/questions/8734. Note that this question concerns precision: accuracy is a different matter altogether. See gis.stackexchange.com/a/8674/664 for the distinction.
– whuber
Nov 8 '18 at 18:10

It depends on the size of the differences between classes. In most applications, saying the 73% prefer option A and 27% prefer option B is perfectly acceptable. But if you're dealing in an election where candidate X has 50.15% of votes and candidate Y has 49.86%, the decimal places are very much necessary.

Of course, you need to take care to make sure that all classes add up to 100%. In my electoral example above, they add up to 100.01%. In that case you might even consider adding a third decimal place.

• I'd also say it depends on your data and your goals. If I were briefing an executive on the sources of customer complaints, no way I'd be talking about fractions of a percent. You also have to consider the margins of error: saying 50.15% of voters prefer candidate X +/- 5% is suspicious. Mar 6 '14 at 15:22
• If there are just two percents, rounding and adding to 100% are completely compatible. See e.g. statweb.stanford.edu/~cgates/PERSI/papers/freedman79.pdf Mar 6 '14 at 15:31
• @NickCox That's a nice find! In case that link goes dead, for future readers the reference is "On Rounding Percentages", Persi Diaconis; David Freedman, Journal of the American Statistical Association, Vol. 74, No. 366. (Jun., 1979), pp. 359-364. Despite the rather broad title, it deals with the probability a table of rounded percentages "correctly" sum to 100. The probability declines to around 3/4 with 3 categories, around 2/3 with 4 categories, and $\sqrt{6/\pi c}$ with $c$ categories Mar 30 '18 at 19:23
• I actually don't think it is necessary to ensure that percentages sum exactly to 100, and wouldn't introduce extra (possibly spurious or distracting) decimal places just to ensure it happens. A little footnote that "Percentages may not sum to 100 due to rounding" should suffice. In fact adding decimal places will not always help, eg the simplest case of having three categories with equal frequencies, then neither 33%+33%+33% nor 33.33%+33.33%+33.33% quite solve the "sum to 100" problem. Mar 30 '18 at 19:34

Different organisations often have conflicting rules for the precision in reporting of results. Ultimately there is a trade-off between when seeing the extra digits is useful, versus cases where unnecessary and excessive precision "can swamp the reader, overcomplicate the story and obscure the message" — a subject explored by Tim Cole (2015) in a piece that I found gave a useful guide to "sensible" precision in reporting, and a comparison of leading style manuals. His advice on percentages was as follows:

Integers, or one decimal place for values under 10%. Values over 90% may need one decimal place if their complement is informative. Use two or more decimal places only if the range of values is less than 0.1%

Examples: 0.1%, 5.3%, 27%, 89%, 99.6%

By "complement" he is referring to cases where one might be interested in the "other lot", e.g. if I tell you 98% of patients in a trial got better, you may well be interested in the 2% who did not, and in that case another decimal place to distinguish whether that "2%" really means "2.4% or "1.6%" would actually be useful.

References

Cole, T. J. (2015). Too many digits: the presentation of numerical data. Archives of disease in childhood, 100(7), 608-609. http://dx.doi.org/10.1136/archdischild-2014-307149

This is a significant figures issue, and is dependent upon the precision of the numbers underlying the percentages. The technically correct number of significant figures is not dependent upon downstream use or the differences between percentage values.

If you're trying to express a percentage describing 5 items out of 7, it would be absurd to claim that it's 71.4285714285% - you simply don't have the precision to back up all those decimal places. When doing division, your answer should have as many significant figures and the fewest number of sig figs in your starting numbers. Here, you only have 1 significant figure, so the percentage should really just be 70%, not even 71%. If you had another example where you want to express 71428 items out of 100000, then you are justified in using more significant figures, all the way out to 71.428%.

Even if you have great precision, it's often preferable to truncate for human readability. Depending on your domain, adding those two extra decimal places may or may not make a difference. You should never over-report significant figures, but you may be justified in under-reporting them if your statistical precision is greater than what's needed for your application.

• Well worth bringing up this issue, but the examples in your second paragraph are rather opaque. I have seven cousins; five are male: in what sense do I lack a precise enough observation to justify reporting this as a percentage to any number of decimal places I please? Mar 30 '18 at 22:27
• I have to agree with @Scortchi. If you know the value of an integer (e.g. number of electrons orbiting an atom, etc) you know it essentially to infinite precision points. We know the proportion of Scortchi's cousins that are male to infinite precision points. What's pragmatic to report is a different issue. (In this case, I think simply saying "5/7" is the best thing.) Nov 8 '18 at 17:52