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"I was told that when I'm rounding, if the number is less than 0.5, I round down, otherwise, I round up. But couldn't that mean more things rounding up than down, since 0.5 is right in the middle, and it gets rounded up?" ~ Quin from Chicago
Hi Quin, before I give you an answer, let me give an example of what you're talking about, to make sure all my readers understand your question.
Suppose you have 8 numbers: 4.1, 3.2, 2.5, 4.5, 5.5, 7.5, 1.6, and 4.9. There are just as many numbers with the tenths place below 0.5 as there are above 0.5, so you might expect that half of them round down, and half of them round up. But that's not what happens. Only two of them round down, and the other six found up. That seems very unbalanced.
Some people might wonder why that even matters. It matters if you have a lot of numbers and you're adding them together.
If you add all of the numbers above you get 33.8 But if you rounded them all, and then added them, you would end up with 36, which is a 6.5% error from the unrounded sum. Now, we don't expect the rounded sum to exactly match the unrounded sum, but this oddity that occurs when you have a bunch of numbers exactly at the midpoint of the rounding makes us wonder (as it made you wonder) if there might be a better way to do this.
It turns out there is an alternative method of rounding which is used in the circumstances described above:
- There are many numbers being added or averaged
- It's not unreasonable to expect that many of the data points will be exactly at the center mark 0.5
Under these circumstances, we can use the following rule for rounding:
If the decimal portion is less than 0.5, we round down, if the decimal portion is more than 0.5, we round up, and if the decimal portion is exactly 0.5, we look at the place value to the left of the five (yes, really, the left!). If it's an odd number, you round up, and if it's an even number, you round down.
For example, our four numbers above that end with a five would round as follows:
2.5 rounds to 2
4.5 rounds to 4
5.5 rounds to 6
7.5 rounds to 8
Another way of saying this is that we always round to the even number in the circumstance where the decimal is exactly 0.5.
So what happens if we do this? Our sum for the values given is 34, which is closer to the 33.8.
There's no guarantee that you won't end up with significant rounding discrepancy (if, by random chance, all your values were less than 0.5, your sum would be way off no matter how you round), but the odds of having large discrepancies decreases if you use this method.
The same method can be used at any place value. If you are rounding 135 to the nearest ten, it would be 140, but 125 would be 120.
Should you use this method of rounding? If you're a student, the answer is: only if your teacher tells you to do it this way!