The two rules for rounding numbers are

- Round your numbers only once (in one step), and
- Round 5's to the nearest even digit -- up or down as needed.

Below I explain why.

In school they usually teach you to round all 5's up to the next digit. For example, 1.45 is rounded to 1.5, 1.65 is rounded to 1.7, 3.225 is rounded to 3.23, etc.

This is wrong because it introduces what we call "systematic error": an error consistently wrong in one direction. In rounding all 5's up, you end up with an average that is too high. ("Random error" goes high or low of the true value randomly, so the average is close to the real value.)

The reason is that 5 is directly in the middle of the digits we round, so we must round it up half the time, and down half the time.

To make this more clear, look at the digits we round to another number: 1, 2, 3, 4 we round down. 6, 7, 8, 9 we round up. (0 we just truncate, since it doesn't affect the previous digit at all.) Notice that 5, if rounded up always, gives us 5 numbers we round up, and 4 we round down. Thus we're rounding up more than rounding down, and we have a positive systematic error. So, in our first list of numbers, we round 1.45 to 1.4, 1.65 to 1.6, and 3.225 to 3.22.

Look at a simple data set. Consider the numbers 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, and 9.5. We want the average value of those numbers after rounding them first. Rounding them as commonly taught gives you 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, and the average of those is = 55/10 = 5.5. (Note that you

**don't round the result, since you've already rounded your numbers once, in the previous step**. Rounding numbers must only be done in one step.)Rounding these numbers to the nearest even digit gives us 0 (0.5 rounded down), 2 (1.5 rounded up), 2 (2.5 rounded down), 4 (3.5 rounded), 4 (4.5 rounded), 6 (etc.), 6, 8, 8, and 10. The average is 50/10 = 5.

The actual average, without rounding, is (0.5 + 1.5 + 2.5 + 3.5 + ... + 9.5)/10 = 50/10 = 5. This is the same result we got when rounding 5's to the nearest even digit.

Rounding all the 5's up to the next digit gives us an error of +0.5/5 = +0.1. That's 10% too high!

If, after rounding all the 5's up to get an average of 5.5 (first example), then rounding that up to 6, we get an even worse error: 20% too high.

Round 5's to the nearest even digit, and round your numbers in only one step. In general, it is best to round only your final answer, as this allows your calculations to carry the most information to the very last step.

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