Rounding Religious Wars

A statement made by Chuck Muhleman in last week's Help Wanted question provoked some interesting comments from many ExcelTips subscribers. Chuck made the statement that, statistically, the value .5 should be round up half the time and down the other half of the time because it is exactly in the middle of two whole values. The analogy was provided that if a tennis ball was balanced on a net, statistically the ball should fall left half of the time and right the other half of the time.

It seems that there are some strong feelings about such statements, even among other statisticians. (All disciplines seem to have their various religious wars where feelings run high.) As one correspondent mentioned, this is "the old 'fences vs. fence posts' problem in counting intervals between numbers." The argument is where something will "fall" when it is situated right on a fencepost. The problem with the tennis ball and net analogy (or fences and fence posts) is that the net in the middle of the court is not the only precise dividing line.

For instance, let's say that the left end of a tennis court has a line marked "4.0" and the other end has a line marked "5.0." This means that the net is marked "4.5." While a tennis ball could balance on the 4.5 mark and fall either way, theoretically the ball could also balance on the line at either end of the court (4.0 and 5.0) and fall either way off of them, as well.

One correspondent expressed the feeling that rounding .5 either up or down (half one way and half the other) is inappropriate because it introduces bias into the data. Consider the situation where you are dealing with one digit to the right of the decimal point: You have numbers 7.0, 7.1, 7.2, etc., all the way through 7.9. When rounding these figures, five values would always round down (7.0 through 7.4), one value could round either way (7.5), and four values would always round up (7.6 through 7.9). In other words, over time 5.5 values would round down and 4.5 values would round up. In a true even application of statistical probability, 5 values should round down and 5 up, but the "waffling" of the center value (7.5) makes a bias in favor of rounding down and against rounding up.

Any discussion of rounding, of course, needs to assume that you are rounding raw values, not previously rounded values. For instance, if a raw value is 14.46 and you round it to 14.5, it would be improper to later round the 14.5 to 15. The correct procedure would be to examine the original 14.46, which should round down, to 14.

So, which theory of rounding is correct? Should 7.5 round up half the time and down half the time, or should it always round up? Microsoft has obviously made its mind up, as it always round 7.5 up (the tennis ball always falls to the right for positive values and to the left for negative values). Does Microsoft's decision mean that always rounding .5 up is correct? Your position in the rounding religious war will determine your answer.