## Abstract

How likely is it that a sum of rounded values is not identical to their rounded sum?

For two random floating point numbers this is obvious: The likelihood is around 25% - that is the percentage of red in this picture:

But it might be somewhat surprising that the likelihood approaches 90% if you round and add more and more numbers:

With seven floating point numbers the likelihood is already larger than 50% that the sum of rounded values is not equal to their rounded sum.

How can you fix this issue? Use RoundToSum.

## Rounded Percentages

Rounded percentages also often fail to add up to 100%.

With two random numbers the issue arises only if both numbers equal 0.5:

But with more random numbers it is similar to the problem stated initially, just with around one number more. Rounded percentages of three arbitrary numbers fail to add up to 1 with a chance of around 25%:

Here with eight random numbers the likelihood is larger than 50% that the rounded percentages do not add up to 1.

Again: with RoundToSum you can fix this.

## Literature

Diaconis, P., & Freedman, D. (13. Juli 2007). (Externer Link!) On Rounding Percentages.

Frederick Mosteller, Cleo Youtz, Douglas Zahn (Demography, 1969, 4, 850-858). The Distribution of Sums of Rounded Percentages.

## Appendix – Monte Carlo Program Codes

Please read my Disclaimer.

```
Option Explicit
Const n = 100
Const runs = 20000
Const bOnlyPositive = True 'Without loss of generality
Sub monte_carlo_add_rounded_values()
'Calculates for 2 to n how likely it is
'that rounding would not alter their sum.
'Example: for 2 numbers there is a 25% chance
'that the sum of their rounded values is not
'equal to their rounded sum.
'Source (EN): https://www.sulprobil.com/rounding_values_alters_their_sum_en/
'Source (DE): https://www.bplumhoff.de/werte_runden_aendert_ihre_summe_de/
'(C) (P) by Bernd Plumhoff 16-Dec-2023 PB V0.3
Dim i As Long
Dim j As Long
Dim k As Long
Dim m As Long
Dim d As Double
Dim s1 As Double
Dim s2 As Double
With Application.WorksheetFunction
Randomize
For i = 2 To n
m = 0
For j = 1 To runs
s1 = 0#
s2 = 0#
For k = 1 To i
If bOnlyPositive Then
d = Rnd()
Else
d = 2# * Rnd() - 1#
End If
s1 = s1 + d
s2 = s2 + .Round(d, 0)
Next k
s1 = .Round(s1, 0)
If s1 <> s2 Then
m = m + 1
End If
Next j
Cells(i, 1) = i
Cells(i, 2) = m / runs
Next i
End With
End Sub
Sub monte_carlo_percentage_sum_of_rounded_values()
'Calculates for 2 to n how likely it is that
'rounding would not alter their percentage sum.
'Example: for 2 numbers there is a 25% chance
'that the sum of their rounded values is not
'equal to their rounded sum.
'Source (EN): https://www.sulprobil.com/rounding_values_alters_their_sum_en/
'Source (DE): https://www.bplumhoff.de/werte_runden_aendert_ihre_summe_de/
'(C) (P) by Bernd Plumhoff 16-Dec-2023 PB V0.2
Dim i As Long
Dim j As Long
Dim k As Long
Dim m As Long
Dim s1 As Double
Dim s2 As Double
With Application.WorksheetFunction
Randomize
For i = 2 To n
m = 0
ReDim e(1 To i) As Double
For j = 1 To runs
s1 = 0#
For k = 1 To i
If bOnlyPositive Then
e(k) = Rnd()
Else
e(k) = 2# * Rnd() - 1#
End If
s1 = s1 + e(k)
Next k
s2 = 0#
For k = 1 To i
e(k) = .Round(1000# * e(k) / s1, 0)
s2 = s2 + e(k)
Next k
If s2 <> 1000# Then
m = m + 1
End If
Next j
Cells(i, 1) = i
Cells(i, 2) = m / runs
Next i
End With
End Sub
```

## Download

Please read my Disclaimer.

How_likely_does_rounding_values_alter_their_rounded_sum.xlsm [43 KB Excel file, open and use at your own risk]