Math Is Fun Forum

Stop the presses!

I discussed it with my roommate, and came to a revelation about something. Instruments of measurement don't truncate the last value they read, it's rounding it. Not by design, but by the nature of the inaccuracy. So, "1.5" on a thermometer won't include 1.5 to 1.59999, it means a value somewhere in the range of .5 to 1.5.

So, I retract pretty much all of my statements. Figuring that out removes a fundamental block in my arguments.

I think the way you round ".5" is dependent on the situation. If you care about consistency (like a programmer would), I guess I'd round up. Other likely factors are minimizing total error or equal representation of the resulting whole numbers.

Yep, I agree with that, of course.

I want to clarify what I consider to be "exactly .5". It has to depend on the situation. With a thermometer, even if it reads .5000000, you should round up, because the temperature could actually be .50000001.

I think I've finally got a good handle on the controversial situation:

There are situations where the decimals are exact figures, and the number of significant digits is actually the degree of acuraccy available. These are absolutely the ONLY cases where I can see how ".5" is a problem, but they do exist.

Lets say I break a pencils into ten pieces each. Then, I grab 5.5 of them. Would it be more accurate to say I grabbed 5 or 6?

From the perspective of the person who sees only the output of the rounding process, I see two main priorities in choosing a rounding method in this situation:
1. making sure that each whole number has an equal chance of "winning".
2. ensuring that the resulting value most closely resembles the original one.

(Note that I don't see "ensuring that rounding up as many times as rounding down" to be a priority. It simply does not matter to the person recieving the data, which I think is the important perspective. In fact, if you were in base 3, then ".0" and ".1" would round down, and ".2" would round up, that's a 66/33% split! But the results most closely match the data, and all round numbers have an equal chance of winning. I think I fell into this trap of thinking earlier.)

And I see three main ways of rounding ".5":
A. Round to the nearest even number.
I hate this one because it blatantly screws up priority #1. I hate it I hate it I hate it.

B. Round randomly up and down, or switch between rounding up and down for each instance of ".5" you come across.
This seems ok, however, this can mess up priority #1. Depending on in what order the numbers chug through the process, you may end up always rounding "1.5" and "2.5" into "2". Now "2" has recieved more than its fair share of the data.

C. Always round up.
This method ensures that you round up half the time and round down half the time. But that's not a priority to whoever recieves the output of our rounding process! So, this factor doesn't count. However, this method can't possibly screw up #1.

".5" isn't any closer to 1 than it is to 0. It's an unfortunate by-product of using base-10! If we used an odd-numbered base, like base 3, then .1 would round down, and .2 would round up, and we'd have no problem. Anyway, rounding up in this situation (".5" in base 10) is no more inaccurate than rounding down, so it's not screwing up priority #2.

So, I like C. You could implement B in a way that priority #1 is still catered to, but it still wouldn't be any better than C, because C satisfies #1 anyway, and satisfies #2 just as much as B does.

So, round up darn it!

I think we agree on what statisticians do. But I still disagree with it. Like I stated in my example, lets say a thermometer reads only to the tenths, but you round to the closest whole number.

2 will get 1.5 to 2.5 ... that's 11 possibilities.
3 will get 2.6 to 3.4 ... that's 9 possibilities.

Each possibility is equally likely to happen. So now we're giving the even numbers recieve a larger span of possibilities. Why is that preferable? After the rounding, the even numbers will be represented as being more frequent than they really are.

But look at your first example, it confuses the issue...

"I think that .50 should not be rounded in either direction because 1/2 is exactly 1/2 away from the adjacent whole numbers.
0       1234      5     6789        0"

Now it IS true that .5 lands right inbetween the adjacent whole numbers, but the important point is that BOTH whole numbers round down. And don't look at them as whole numbers, look at them as their decimal equivalents... they're really ".0", and the ".0" happens as frequently as ".1", ".2", up to ".9" will. There is rarely the case when these freuquencies are not equal, especially in the cases of significant digits / precision.

And something about your second example is resting uneasily with me, but I can't put my finger on it yet.