Rounding is more difficult than first appears. It appears straight-forward. To round a number you decide how many decimal places or significant figures you need then you look one digit further to see whether the final digit stays the same or goes up. Presto – there is rounding in a nutshell. Yet my university students struggle with rounding to a surprising degree. I did a Youtube search on rounding for a video to help them, but to no avail.
I wrote a script for such a video. I’m afraid it won’t be appearing any time soon as I now have to work for my living (as opposed to being an academic ;) ) but the exercise was interesting. What I realised is that rounding is about communication. It has nothing to do with mathematics and everything to do with expression.
The problem is one of judgment. There is no black and white answer. Firstly the rule is that you DON’T round during a calculation, and you DO round at the end. Too many students manage to either round early in the calculation and increase error in their calculations, or keep every decimal place, forever! Whatever number of decimal places their calculator gives them, that is what is reported.
Even at honours level we find students reporting with spurious precision: “the savings from this approach will be $45,923.” My colleague (the one who has a thing about titles) likes to ask “Are you sure about that? Could it be $45,922?” She is, of course, not serious, but trying to point out the unnecessary non-zero digits in their estimation.
But then my mathematically-minded colleague points out that there is nothing inherently wrong with saying $45,923. In fact by rounding to $46,000 they are moving away from the central value of their estimate, and making it worse. And mathematically that is true. But rounding isn’t about the mathematics – it is about communication. When we state $46,000 everyone knows we don’t mean exactly $46,000. There is an implied level of variation of about $500 either way. Or it could be $250 either way, because maybe they would round $45,523 to $45,500. It is this horrible greyness that abstract mathematicians have escaped, which pervades real-life subjects like statistics and operations research.
This is where the judgment call comes in. In science there are rules about the number of decimal places used, depending on the number of decimal places in the values or measurements used in the calculation. But in statistics the rules are fuzzier. And when we are dealing with money there are certain unwritten rules. In New Zealand we usually state money values to either two decimal places or none. Even though we have divested ourselves of coinage smaller than 10 cents, most prices are given to the nearest cent. Some restaurants are starting to give prices to one decimal place, but that is the exception.
Here is an example that intrigues me. On our bathroom wall we have a card teaching about CPR. It was obviously originally written in “Imperial land”. The distance of chest compression is given as 2.54cm. Do you think it used to say one inch? And did they really mean exactly one inch? I doubt it. This should have been changed to 2cm or 3cm. Maybe 2.5cm if they really think someone can estimate that accurately when trying to keep their loved-one alive by pressing on their chest.
These days with spreadsheets, the mechanical aspect of rounding is simpler. We teach the students to use the little decimal place button in Excel to do the rounding. That way the number will be correctly rounded. We also try to teach them that the underlying number in the cell remains at a higher level of precision, and that only the appearance has changed.
However, the spreadsheet does not remove the need for the decision about how many decimal places to use. What are we communicating? The role of a confidence interval is to express an estimation, so stating a confidence interval to high precision is laughable. Stating, for example, that we are 95% confident that the mean of the population lies between 22.478 and 35.721 indicates lack of understanding of the nature of a confidence interval. I would use one decimal place at most and give the confidence interval as (22.5, 35.7). Somehow to use no decimal places seems cavalier, though really it would be more sensible.
Recently I received a report about student progress, given in percentages to three decimal places. To me this undermined the validity of the report, as the person who wrote it clearly did not know about sensible rounding. This made me wonder about the validity of the rest of the report. Rounding is about communication, and this lack of sensible rounding communicated a message I am sure the author did not intend.