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Rounding Religious Wars

A statement made by an ExcelTips subscriber (Chuck) in answer to a Help Wanted question provoked some interesting comments from many other 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.

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Comments for this tip:

Tom Bates    19 May 2016, 16:26
Thanks Rick Rothstein for that tip. That will come in very handy!
Rick Rothstein    18 May 2016, 05:32
Tom Bates gets what I was trying to say in my first post. By the way Tom, every function in VB that uses rounding internally (such as CLng) uses what many call "Banker's Rounding" (another name for the rounding to even numbers method) with one exception... the Format function which is the only VB function that performs what I call normal rounding (5s always round up). So you can use Format to perform your rounding in VB without calling out to the worksheet's ROUND function. For example,

Num = 2.45
MsgBox Format(Num, "0.0")
Tom Bates    17 May 2016, 08:51
Consider this series:

tenths =ROUND() CLng()
1.2 1.0 1
1.3 1.0 1
1.4 1.0 1
1.5 2.0 2
1.6 2.0 2
1.7 2.0 2
1.8 2.0 2
1.9 2.0 2
2.0 2.0 2
2.1 2.0 2
2.2 2.0 2
2.3 2.0 2
2.4 2.0 2
2.5 3.0 2
2.6 3.0 3
2.7 3.0 3
2.8 3.0 3
2.9 3.0 3
3.0 3.0 3
3.1 3.0 3
3.2 3.0 3
3.3 3.0 3
3.4 3.0 3
3.5 4.0 4
3.6 4.0 4
3.7 4.0 4
3.8 4.0 4

From this list, we see that there are 10 2.0 values in column 2, and 10 3.0 values, because the ROUND function is correctly rounding .5 up.

In contrast, the CLng function rounds incorrectly, because the results are obviously very skewed towards even numbers: we see more 2's (11) than 3's (9).

Thanks to stas for pointing that out; henceforth, I will be using the spreadsheet function ROUND rather than allowing VBA to perform the rounding incorrectly.
Tom Bates    17 May 2016, 08:30
The fact that rounding 7.0 yields 7.0 does not mean that it has not been rounded. It simply means that the result of rounding yields the same number.

If you consider rounding to a multiple of 10, 70 *remains* 70, while 71 *becomes* 70. It may be easier to recognize that 70 to 79 are "the seventies"; 70 to 74 are the lower half, 75 to 79 the upper half, exactly even. 80 is clearly not part of the 70's.

So .49999999999999 rounds down, .5 rounds up. Always (mathematically speaking).
Scott Renz    16 May 2016, 18:42
In the example 7.0, 7.1, 7.2, etc., 7.0 does not get rounded. It keeps its same value. So, only the 7.1, 7.2, 7.3, and 7.4 get rounded down.
Rick H.    16 May 2016, 13:47
Wow. Did I make this more complicated than necessary?
Trying to stay away from VBA, I worked up this formula to round cell C16 to two places and rounding even when the third and final digit was a 5:
If it matters, I'm still using Excel 2000, Win7.
stas    16 May 2016, 10:15
The more striking is a difference between "round" worksheet function and a "round" in VBA. Worksheet function always round 0.5 up. VBA "round" rounds 0.5 to the even value.

worksheet round VBA round
0.5 1 0
1.5 2 2
2.5 3 2
3.5 4 4
4.5 5 4
5.5 6 6
6.5 7 6
7.5 8 8
8.5 9 8
9.5 10 10
10.5 11 10
11.5 12 12
12.5 13 12
13.5 14 14
14.5 15 14
15.5 16 16
16.5 17 16

  VBA round was from
For i = 1 To 17
    Cells(i, 3) = Round(Cells(i, 1), 0)
Next i

 I Use Excel 2013 with Win7.
Rick H.    16 May 2016, 09:24
Why are we including 7.0 in the series of values without including 8.0?
7.0 is not rounded down. The decimal is simply dropped. 7.0 = 7
This leaves 7.1 through 7.4 (four values) always rounding down, and 7.6 through 7.9 (four values) always rounding up.
Are the values in this scenario only between 7.0 and 8.0?
I was taught to round even. This takes care of skewing the data, you don't have to keep track of which direction the last value was rounded, and subsequent halving isn't left with another .5 to deal with.
If our scenario includes values from a minimum of 0 to a maximum of 10, and we always round the .5 values up, the average is skewed. Rounding even corrects this issue.
Raw values: .5+1.5+...8.5+9.5=50, Avg.=5
Rounding up: 1+2+...9+10=55, Avg.=5.5 which rounds to 6.
Rounding even: 0+2+...8+10=50; Avg.=5
Roger    14 May 2016, 08:17
I was just discussing this the other day with a client. I told her that I used to teach children in Australia and this is how I explained it to them.... "My uncle used to live in England and he wanted to swim to France across the English Channel. He was VERY keen to do this. So off he went and swam and swam and swam. When he got exactly half way, he was so tired and knew he could not make it - so he swam back to England. Hands flew up and asked why he did not go on because it was the same distance going to France as swimming back. I said, 'That is why we round up rather than down'."
Rick Rothstein    14 May 2016, 05:54
No question about it, .5 should always be rounded up... and the reason is simplicity itself. Let's use the 7.5 for our example. The lower interval is 7.0 to 7.4 and the upper interval is 7.5 to 7.9 (8 belongs to the next interval and does not figure into the decision). Those two intervals contain exactly the same number of values... the lower interval rounds down and the upper interval rounds up... and since 7.5 belongs to the upper interval, it should always round up just like the other members of its interval. Simple, right?

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