Why is 5 always rounded up?

Help!

I got into a big argument with my boss today about whether or not 5 should be rounded up or down. He says it doen’t matter if youy round up or down, as long as everyone always does it the same way. Others say to always round up.

I was taught that if 5 is preceeded by an even number, round down (1.25 -> 1.2). If it’s preceeded by an odd number, round up (1.15 -> 1.2). The theory is that since 5 is exactly halfway between 0 and 10, it should be rounded up half the time and down the other half, so in the long run it will balance out. Otherwise, when rounding a large number of numbers the result will be biased if 5 is always rounded one way.

Which is correct, and why?

The issue isn’t whether 5 is halfway between 0 and 0, because we don’t have zeros at both ends of our numbering system. We only have 10 digits, 0 through 9, and 5 is more than halfway through that sequence, so it rounds up. What makes this confusing for some people is that rounding up means going to the next sequence, which starts with a 0 again.

Both are accepted forms of rounding, but your method is more accurate. It’s commonly referred to as “Banker’s rounding” and it’s purpose is to neutralize the results over a large data sample. It’s important to note that you don’t actually round down on even numbers, you round towards zero. This makes a difference when dealing with negative numbers. Unfortunately, the less accurate method of always rounding up is much more common in actual use.

If neutralizing the results isn’t important, your boss is correct when stating that it’s more important that everyone uses the same method. Otherwise the results won’t always match.

Assuming we need to round to cents and that the rounding is performed on each operand before the operation:

Banker’s rounding – 1.125 + 1.125 = 1.24

“Traditional” rounding – 1.125 + 1.125 = 1.26

This reminds me of the story I once heard (may be a Urban Legend):

A computer programmer who worked at the Social Security Adminstration would round DOWN any fractions of a cent computed for benefit checks and have the remainder direct-deposited into his own bank account. With millions of checks going out every month, this amounted to a substantial sum. He was eventually caught and convicted.

Haven’t thought about this for awhile. I guess I need to check out Snopes.

I still love it, true or not! Sometimes human ingenuity just boggles the mind.

Searched on Snopes and the SD columns. Nothing.

No promising thread titles here, either…

So, what’s the Straight Dope? Should I post this story in GQ?

Actually, you’re thinking of the trick used by Richard Pryor’s character in Superman III.

Daemon, I’ve definately read that before. I’m not sure of the specifics.

DMC, I think you mean:

Banker’s rounding – 1.125 + 1.125 = 2.24

“Traditional” rounding – 1.125 + 1.125 = 2.26

Ell, my bad.

That’s why I use computers to do all of my calculating for me. :slight_smile:

You’re right Saltire - the character in that film did do something like the rounding scam, but the story/legend precedes the release of the film by many years, I suspect it’s inclusion in the film was inspired by the story.

dammit, should preview first

Ell’s example, after correction, shows why it is important not to round intermediate products and dividends (at least anymore than you have to given the physical reality of computers).

1.125+1.125=2.25

which does not need rounding and is precisely correct.

The problem grows enormously when two nearly equal numbers are subtracted and the result is then used in the denominator of a fraction. If one rounds first, there is a high risk of dividing by zero (or a “near zero” which is not actually equal to the true result) which is never a pretty sight.

Basically, save your rounding for last!

Good point - in most cases it’s a good idea to round at the highest level realistically possible, so on an invoice with VAT (sales tax), I would calculate the total tax on the total of all applicable lines, then round, rather than caclulate and round line-by line then add it all up (although some people do it this way)

Usually there are other non-zero digits past the rounding point. If there are other non-zero digits past the 5, you are not half-way between you are past the mid-point. Therefore, you should round 5’s up. Only if you are using exact numbers and the 5 is the last non-zero digit would bankers’ rounding make sense.

Just to make it more confusing, I was taught by different teachers in different schools to round five up if the digit preceding it is odd and down if the preceding digit is even. 1.125 would round to 1.12 and 1.375 would round to 1.38.

What is being called here “bankers’ rounding” is not rounding at all. That process is called “truncation” and is simply the lopping off of all unwanted digits, regardless of the effect on any resultant computations. You will probably notice if you look carefully that money you owe the bank gets rounded, but money the bank owes you gets truncated.

That way, the bank gets to keep all the “fractions of a cent.” :slight_smile:

First, your subject question assumes that 5 is “always” rounded up.

We’ve established that this is not the case. 5 is sometimes rounded down. This might beg the question, “Why is 5 usually rounded up?”. I would say that whenever someone is applying the rule of rounding up 5, it is usually in a context where it doesn’t make a lot of friggin’ difference.

Most dunderheads who are repeatedly rounding up 5 probably aren’t doing any striking calculations. It’s sort of the math world’s conspiracy against - while acknowledging - ignorance. “Hey, should we bother explaining a rule of rounding to everyone - like the way bankers round - or should we just tell everyone to round 5 up?” Response: “The dolts of the world can barely manage the rounding rule of rounding 5 up, so we’ll just skip any more details and leave it at that.”

0,1,2,3,4 /5 /,6,7,8,9

Wanna re-think that Knead?

There are industry standards for rounding.

For testing ASTM E29 is used. If the number ends in a 5 you round, up or down, to the even digit. Both 1.15 and 1.25 round to 1.2

Sorry about the inaccurate thread title. I was in a rush. Should have been something like: “How do you round 5?”

I appreciate that in most cases it really doesn’t matter how you round, but you know how geeks get. We gotta know which way is correct assuming it does matter.

Anyway, I like starfish’s solution: “If the number ends in a 5 you round, up or down, to the even digit. Both 1.15 and 1.25 round to 1.2”.

Let’s say we have to round a bunch of numbers and then add them all up. Which rounding method will be more accurate? I’m guessing always rounding up or always rounding down will skew the sum up or down, respectively. (By how much? Maybe a lot if there are a lot of 5’s to round.) starfish’s rounding, over a large enough random sample, will not skew the result by more than half of a significant digit, giving the same result as rounding after summing.

But why is a particular method better? Are there any mathematicians willing to explain it to an engineer? starfish, where do I get a copy of ASTM E29? I need to convince an engineer who is positive I’m wrong.

Yuck:

You are soooo correct in your statement, but unfortunately it’s not always easy to do. I use banker’s rounding in an application that I wrote to handle student loan processing. Their database stores all dollar figures with only two decimal places (I tried to talk them into changing that, to no avail). Most of the fees are calculated as a percentage of the loan amount, hence requiring rounding. These are then stored in the database. At various points in the process, these numbers are used in further calculations, thus causing even more problems. Needless to say, my more accurate version produced numbers that were not matched by the auditors, the insurers, nor the bank, all of whom rounded up on 5.

The best way in that situation would have been to calculate the later numbers by recalculating the fees, and using the much longer number that is stored in memory instead of the database, but that wasn’t an option in their case.

KneadToKnow:

How is Banker’s rounding the same as truncation?

Banker’s rounding – 1.135 = 1.14

Truncation – 1.135 = 1.13

If you mean that not all banks use this type of rounding, I’ll agree. On a related note, if anyone has read Regulation Z, you’d understand why all of the confusion exists in different financial institutions. That is one messed up document. :slight_smile: