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:

Two Values

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

Rounding Values alters their Sum

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:

Two Values

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%:

Percentage Values not adding up to 1

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]