A colleague of mine posted a query about rounding numbers and I am not at all sure about my reply which I've spoilered below. Am I even correct? And can anyone provide a concise easily understandable explanation? I am dissatisfied with my argument being based on an example rather than a principle.
Thanks in advance for your help.
I’m no mathematician, but I don’t understand why there needs to be this augmentation to the basic rounding rules that we all learned in first or second grade. If I’m given the instruction to round a given number to two significant figures, how does rounding a “5” up or down 50 % of the time making the rounding process more even?
Using the “original” method, I would round the value 0.220 to 0.22, 0.221 to 0.22, 0.222 to 0.22, 0.223 to 0.22, 0.224 to 0.22, 0.225 to 0.23, 0.226 to 0.23, 0.227 to 0.23, 0.228 to 0.23, and 0.229 to 0.23. That gives my five values that are round to 0.22 and five values rounded to 0.23. If I look at all of the possible values of 0.22 out to the forth decimal place, I would have 50 values that would be rounded to 0.22 and 50 values that would be rounded to 0.22, and this goes on and on in powers of ten for infinity. Are we forgetting that “0” is a number too? In the field of measurement science, the value 0.220 is not the same as 0.22.
So my question is, why is this method created in the first place and why should anyone feel it is necessary to use it?
If we look at this question in terms of the 'distance moved' in rounding the number then we might get a very different result : Assume a flat distribution of results.
For the 5 results that would be rounded down:
.22 -.220 = 0
.22 - .221 = -.001
.22 - .222 = -.002
.22 - .223 = - .003
.22 - .224 = - .004
And potentially the total distance moved while rounding down is -.010.
Similarly for rounding up :
.23 - .225 = .005 \\ .23 - 226 = .004 \\ .23 -.227 = .003 \\ .23-.228 = .002 \\ .23 -.229 = .001
And the total distance moved while rounding up would be +.015. This would bias the average of your total population higher.
But I'm no expert. It might be nice if someone better qualified weighed in with an argument.