Saturday, October 10, 2009

Rounding Up: A Roundup

The Dow Jones Industrial Average closed Oct. 9, 2009, at 9864.94. If you want to express that as a whole number, you just lop off everything after the decimal point, right? Well, of course not. It's closer to 9865 than to 9864, and so you'd round it to 9865. Also Friday, the Standard & Poor's 500 index closed at 1071.49. You'd round that to 1071. But if it had risen one more hundredth of a point, to land midway between two whole numbers, for rounding purposes that would equal an entire point -- 1071.50 would round to 1072. Tie goes to the runner, or something like that.

Now then. The half-rounds-up concept can get you in trouble. You'd probably never lop just one of the decimal digits off a stock index, but let's say a reporter was figuring out how much one of those indexes had risen since a certain date, and the answer came out to 14.45 percent. That's a lot of not-so-significant detail, and so maybe the reporter's assignment editor would round that up to 14.5 percent. And then the story comes to you, the copy editor, and you think, wow, that's a lot of not-so-significant detail, so I'll just make that a whole number. The half rounds up, and so you make it 15 percent.

You've now inserted an error.

When you go to round a number up, you'd better make darn sure that you're not rounding an already-rounded number.

In a related caution, take a look at the following snippet from a spreadsheet:

Would you believe that both sums, in their own way, are correct?

Here are the same two columns and sums, only expanded to three decimal places.

In the first column, Excel was showing the rounded numbers but adding the raw numbers. In the second column, Excel was adding up only the rounded versions. Lesson No. 1: It's fine to round the result of an equation, but do not round the numbers you use to get that result. Lesson No. 2: If you're diligently checking the numbers in a report, or in somebody else's graphic, and the answer doesn't add up, consider that you may not have enough information to judge whether the math was correct.


nvossbrink said...

Lesson 3. Don't forget what you learned about significant figures in science class. The answer of any calculation can only be accurate to the number of significant figures in the measurements.

bensrib said...

That's exactly what I was thinking as I read that! I recently edited an explanation of significant figures, so it's fresh in my mind. Your answer can't be more exact than the measurements you begin with.

Tina Russell said...

Oooh! Also, remember that the reason why 0.50 rounds to 1 is because 0.00 through 0.49—fifty separate numbers—round to 0. So, it makes sense that 0.50 through 1.00—another fifty separate numbers—would round to 1. Class dismissed!

Seriously, your post is all correct and great, I just thought I’d make it a little more well-rounded ;)

Nixta said...

Wait, don't you actually need an extra step in there for this mistake (unless I'm being even dimmer than usual)?

No-one should round 14.449 to 14.5 directly because you take the next significant figure and use that for rounding.

That is, to round to 1 decimal place, you would take 2 decimal places (lopped, yes lopped) and use the 2nd place to decide the new 1st, so you'd round 14.449 to 14.4, not 14.5.

However, if you had someone else in-between *they* might go 3 decimal places to 2 and round 14.449 to 14.45, then the assignment editor would go 2 decimal places to 1 and round 14.45 to 14.5 (it's a game of Telephone with numbers!).

Then you're in a mess. I take it, since you didn't mention it, that there isn't a recognized notation to indicate a number's been rounded already? E.g. "... 14.5[14.449] percent ..."?

Bill said...

Good catch, Nixta. I should have made it 14.45 becoming 14.5 becoming 15.

Dominic Sayers said...

@Tina Russell: 0.50 through 1.00 is fifty-one separate numbers.

I think you meant 0.50 through 0.99.


Typical Internet Pedant

onemore4gsus said...

So that's why an IRS Quarterly 941 never matches to the penny?