How to make rounded percentages add up to 100%

Probably the "best" way to do this (quoted since "best" is a subjective term) is to keep a running (non-integral) tally of where you are, and round that value.

Then use that along with the history to work out what value should be used. For example, using the values you gave:

Value      CumulValue  CumulRounded  PrevBaseline  Need
---------  ----------  ------------  ------------  ----
                                  0
13.626332   13.626332            14             0    14 ( 14 -  0)
47.989636   61.615968            62            14    48 ( 62 - 14)
 9.596008   71.211976            71            62     9 ( 71 - 62)
28.788024  100.000000           100            71    29 (100 - 71)
                                                    ---
                                                    100

At each stage, you don't round the number itself. Instead, you round the accumulated value and work out the best integer that reaches that value from the previous baseline - that baseline is the cumulative value (rounded) of the previous row.

This works because you're not losing information at each stage but rather using the information more intelligently. The 'correct' rounded values are in the final column and you can see that they sum to 100.

You can see the difference between this and blindly rounding each value, in the third value above. While 9.596008 would normally round up to 10, the accumulated 71.211976 correctly rounds down to 71 - this means that only 9 is needed to add to the previous baseline of 62.

---

This also works for "problematic" sequence like three roughly-1/3 values, where one of them should be rounded up:

Value      CumulValue  CumulRounded  PrevBaseline  Need
---------  ----------  ------------  ------------  ----
                                  0
33.333333   33.333333            33             0    33 ( 33 -  0)
33.333333   66.666666            67            33    34 ( 67 - 33)
33.333333   99.999999           100            67    33 (100 - 67)
                                                    ---
                                                    100

There are many ways to do just this, provided you are not concerned about reliance on the original decimal data.

The first and perhaps most popular method would be the Largest Remainder Method

Which is basically:

1. Rounding everything down
2. Getting the difference in sum and 100
3. Distributing the difference by adding 1 to items in decreasing order of their decimal parts

In your case, it would go like this:

13.626332%
47.989636%
 9.596008%
28.788024%

If you take the integer parts, you get

13
47
 9
28

which adds up to 97, and you want to add three more. Now, you look at the decimal parts, which are

.626332%
.989636%
.596008%
.788024%

and take the largest ones until the total reaches 100. So you would get:

14
48
 9
29

Alternatively, you can simply choose to show one decimal place instead of integer values. So the numbers would be 48.3 and 23.9 etc. This would drop the variance from 100 by a lot.