How to make rounded percentages add up to 100

Question

Consider the four percentages below  represented as float numbers       13 626332      47 989636       9 596008      28 788024     -----------    100 000000    I need to represent these percentages as whole numbers  If I simply use Math round    I end up with a total of 101     14   48   10   29   101   If I use parseInt    I end up with a total of 97     13   47   9   28   97   What s a good algorithm to represent any number of  percentages as whole numbers while still maintaining a total of 100      Edit  After reading some of the comments and answers  there are clearly many ways to go about solving this   In my mind  to remain true to the numbers  the  right  result is the one that minimizes the overall error  defined by how much error rounding would introduce relative to the actual value           value  rounded     error               decision    ----------------------------------------------------     13 626332       14      2 7           round up  14      47 989636       48      0 0           round up  48       9 596008       10      4 0     don t round up   9      28 788024       29      2 7           round up  29    In case of a tie  3 33  3 33  3 33  an arbitrary decision can be made  e g  3  4  3

User · Answer

If you are rounding it there is no good way to get it exactly the same in all case.

You can take the decimal part of the N percentages you have (in the example you gave it is 4).

Add the decimal parts. In your example you have total of fractional part = 3.

Ceil the 3 numbers with highest fractions and floor the rest.

(Sorry for the edits)

User · Answer

This is a case for banker s rounding  aka  round half-even   It is supported by BigDecimal  Its purpose is to ensure that rounding balances out  i e  doesn t favour either the bank orthe customer

User · Answer

I think the following will achieve what you are after  function func  orig  target          var i   orig length  j   0  total   0  change  newVals       next  factor1  factor2  len   orig length  marginOfErrors               map original values to new array     while  i--             total    newVals i    Math round  orig i                change   total  lt  target   1   -1       while  total     target                 Iterate through values and select the one that once changed will introduce            the least margin of error in terms of itself  e g  Incrementing 10 by 1            would mean an error of 10  in relation to the value itself          for  i   0  i  lt  len  i                    next   i     len - 1   0   i   1               factor2   errorFactor  orig next   newVals next    change                factor1   errorFactor  orig i   newVals i    change                 if   factor1  gt  factor2                     j   next                                    newVals j     change          total    change              for  i   0  i  lt  len  i       marginOfErrors i    newVals i   amp  amp  Math abs  orig i  - newVals i      orig i             Math round   causes some problems as it is difficult to know at the beginning        whether numbers should have been rounded up or down to reduce total margin of error          This section of code increments and decrements values by 1 to find the number        combination with least margin of error      for  i   0  i  lt  len  i               for  j   0  j  lt  len  j                   if  j     i   continue               var roundUpFactor   errorFactor  orig i   newVals i    1     errorFactor  orig j   newVals j  - 1                var roundDownFactor   errorFactor  orig i   newVals i  - 1    errorFactor  orig j   newVals j    1                var sumMargin   marginOfErrors i    marginOfErrors j                if  roundUpFactor  lt  sumMargin                     newVals i    newVals i    1                  newVals j    newVals j  - 1                  marginOfErrors i    newVals i   amp  amp  Math abs  orig i  - newVals i      orig i                   marginOfErrors j    newVals j   amp  amp  Math abs  orig j  - newVals j      orig j                              if  roundDownFactor  lt  sumMargin                      newVals i    newVals i  - 1                  newVals j    newVals j    1                  marginOfErrors i    newVals i   amp  amp  Math abs  orig i  - newVals i      orig i                   marginOfErrors j    newVals j   amp  amp  Math abs  orig j  - newVals j      orig j                                       function errorFactor  oldNum  newNum             return Math abs  oldNum - newNum     oldNum             return newVals      func  16 666  16 666  16 666  16 666  16 666  16 666   100        gt   16  16  17  17  17  17  func  33 333  33 333  33 333   100        gt   34  33  33  func  33 3  33 3  33 3  0 1   100        gt   34  33  33  0   func  13 25  47 25  11 25  28 25   100         gt   13  48  11  28  func   25 5  25 5  25 5  23 5   100         gt   25  25  26  24    One last thing  I ran the function using the numbers originally given in the question to compare to the desired output  func  13 626332  47 989636  9 596008  28 788024   100        gt   48  29  13  10    This was different to what the question wanted      48  29  14  9   I couldn t understand this until I looked at the total margin of error  -------------------------------------------------   original    question     diff   mine     diff   -------------------------------------------------   13 626332   14         2 74     13     4 5        47 989636   48         0 02     48     0 02       9 596008    9          6 2      10     4 2        28 788024   29         0 7      29     0 7      -------------------------------------------------   Totals      100        9 66     100    9 43     -------------------------------------------------   Essentially  the result from my function actually introduces the least amount of error   Fiddle here

User · Answer

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    Rounding everything down Getting the difference in sum and 100 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

User · Answer

I once wrote an unround tool  to find the minimal perturbation to a set of numbers to match a goal  It was a different problem  but one could in theory use a similar idea here  In this case  we have a set of choices   Thus for the first element  we can either round it up to 14  or down to 13  The cost  in a binary integer programming sense  of doing so is less for the round up than the round down  because the round down requires we move that value a larger distance  Similarly  we can round each number up or down  so there are a total of 16 choices we must choose from     13 626332   47 989636    9 596008   28 788024 -----------  100 000000   I d normally solve the general problem in MATLAB  here using bintprog  a binary integer programming tool  but there are only a few choices to be tested  so it is easy enough with simple loops to test out each of the 16 alternatives  For example  suppose we were to round this set as    Original      Rounded   Absolute error    13 626           13          0 62633     47 99           48          0 01036     9 596           10          0 40399    28 788           29          0 21198 ---------------------------------------   100 000          100          1 25266   The total absolute error made is 1 25266  It can be reduced slightly by the following alternative rounding    Original      Rounded   Absolute error    13 626           14          0 37367     47 99           48          0 01036     9 596            9          0 59601    28 788           29          0 21198 ---------------------------------------   100 000          100          1 19202   In fact  this will be the optimal solution in terms of the absolute error  Of course  if there were 20 terms  the search space will be of size 2 20   1048576  For 30 or 40 terms  that space will be of significant size  In that case  you would need to use a tool that can efficiently search the space  perhaps using a branch and bound scheme

User · Answer

DO NOT sum the rounded numbers   You re going to have inaccurate results   The total could be off significantly depending on the number of terms and the distribution of fractional parts     Display the rounded numbers but sum the actual values   Depending on how you re presenting the numbers  the actual way to do that would vary   That way you get    14  48  10  29     100   Any way you go you re going to have discrepancy   There s no way in your example to show numbers that add up to 100 without  rounding  one value the wrong way  least error would be changing 9 596 to 9   EDIT  You need to choose between one of the following    Accuracy of the items Accuracy of the sum  if you re summing rounded values  Consistency between the rounded items and the rounded sum    Most of the time when dealing with percentages  3 is the best option because it s more obvious when the total equals 101  than when the individual items don t total to 100  and you keep the individual items accurate    Rounding  9 596 to 9 is inaccurate in my opinion   To explain this I sometimes add a footnote that explains that the individual values are rounded and may not total 100  - anyone that understands rounding should be able to understand that explanation

User · Answer

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

User · Answer

The goal of rounding is to generate the least amount of error  When you re rounding a single value  that process is simple and straightforward and most people understand it easily  When you re rounding multiple numbers at the same time  the process gets trickier - you must define how the errors are going to combine  i e  what must be minimized   The well-voted answer by Varun Vohra minimizes the sum of the absolute errors  and it s very simple to implement  However there are edge cases it does not handle - what should be the result of rounding 24 25  23 25  27 25  25 25  One of those needs to be rounded up instead of down  You would probably just arbitrarily pick the first or last one in the list   Perhaps it s better to use the relative error instead of the absolute error  Rounding 23 25 up to 24 changes it by 3 2  while rounding 27 25 up to 28 only changes it by 2 8   Now there s a clear winner   It s possible to tweak this even further  One common technique is to square each error  so that large errors count disproportionately more than small ones  I d also use a non-linear divisor to get the relative error - it doesn t seem right that an error at 1  is 99 times more important than an error at 99   In the code below I ve used the square root   The complete algorithm is as follows    Sum the percentages after rounding them all down  and subtract from 100  This tells you how many of those percentages must be rounded up instead  Generate two error scores for each percentage  one when when rounded down and one when rounded up  Take the difference between the two  Sort the error differences produced above  For the number of percentages that need to be rounded up  take an item from the sorted list and increment the rounded down percentage by 1    You may still have more than one combination with the same error sum  for example 33 3333333  33 3333333  33 3333333  This is unavoidable  and the result will be completely arbitrary  The code I give below prefers to round up the values on the left   Putting it all together in Python looks like this   def error gen actual  rounded       divisor   sqrt 1 0 if actual  lt  1 0 else actual      return abs rounded - actual     2   divisor  def round to 100 percents       if not isclose sum percents   100           raise ValueError     n   len percents      rounded    int x  for x in percents      up count   100 - sum rounded      errors     error gen percents i   rounded i    1  - error gen percents i   rounded i    i  for i in range n       rank   sorted errors      for i in range up count           rounded rank i  1      1     return rounded   gt  gt  gt  round to 100  13 626332  47 989636  9 596008  28 788024    14  48  9  29   gt  gt  gt  round to 100  33 3333333  33 3333333  33 3333333    34  33  33   gt  gt  gt  round to 100  24 25  23 25  27 25  25 25    24  23  28  25   gt  gt  gt  round to 100  1 25  2 25  3 25  4 25  89 0    1  2  3  4  90    As you can see with that last example  this algorithm is still capable of delivering non-intuitive results  Even though 89 0 needs no rounding whatsoever  one of the values in that list needed to be rounded up  the lowest relative error results from rounding up that large value rather than the much smaller alternatives   This answer originally advocated going through every possible combination of round up round down  but as pointed out in the comments a simpler method works better  The algorithm and code reflect that simplification

User · Answer

Here s a Ruby gem that implements the Largest Remainder method  https   github com jethroo lare round To use  a    Array new 3   BigDecimal  0 3334         gt     lt BigDecimal 887b6c8  0 3334E0  9 18  gt     lt BigDecimal 887b600  0 3334E0  9 18  gt     lt BigDecimal 887b4c0  0 3334E0  9 18  gt   a   LareRound round a 2      gt     lt BigDecimal 8867330  0 34E0  9 36  gt     lt BigDecimal 8867290  0 33E0  9 36  gt     lt BigDecimal 88671f0  0 33E0  9 36  gt   a reduce     to f     gt  1 0

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I have implemented the method from Varun Vohra s answer here for both lists and dicts   import math import numbers import operator import itertools   def round list percentages number list               Takes a list where all values are numbers that add up to 100      and rounds them off to integers while still retaining a sum of 100       A total value sum that rounds to 100 00 with two decimals is acceptable      This ensures that all input where the values are calculated with  fraction   total      and the sum of all fractions equal the total  should pass                Check input     if not all isinstance i  numbers Number  for i in number list           raise ValueError  All values of the list must be a number          Generate a key for each value     key generator   itertools count       value dict    next key generator   value for value in number list      return round dictionary percentages value dict  values     def round dictionary percentages dictionary               Takes a dictionary where all values are numbers that add up to 100      and rounds them off to integers while still retaining a sum of 100       A total value sum that rounds to 100 00 with two decimals is acceptable      This ensures that all input where the values are calculated with  fraction   total      and the sum of all fractions equal the total  should pass                Check input       Only allow numbers     if not all isinstance i  numbers Number  for i in dictionary values             raise ValueError  All values of the dictionary must be a number         Make sure the sum is close enough to 100       Round value sum to 2 decimals to avoid floating point representation errors     value sum   round sum dictionary values     2      if not value sum    100          raise ValueError  The sum of the values must be 100          Initial floored results       Does not add up to 100  so we need to add something     result    key  int math floor value   for key  value in dictionary items           Remainders for each key     result remainders    key  value   1 for key  value in dictionary items          Keys sorted by remainder  biggest first      sorted keys    key for key  value in sorted result remainders items    key operator itemgetter 1   reverse True          Otherwise add missing values up to 100       One cycle is enough  since flooring removes a max value of  lt  1 per item        i e  this loop should always break before going through the whole list     for key in sorted keys          if sum result values       100              break         result key     1        Return     return result

User · Answer

Here s a simpler Python implementation of  varun-vohra answer   def apportion pcts pcts  total       proportions    total    pct   100  for pct in pcts      apportions    math floor p  for p in proportions      remainder   total - sum apportions      remainders     i  p - math floor p   for  i  p  in enumerate proportions       remainders sort key operator itemgetter 1   reverse True      for  i     in itertools cycle remainders           if remainder    0              break         else              apportions i     1             remainder -  1     return apportions   You need math  itertools  operator

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Since none of the answers here seem to solve it properly  here s my semi-obfuscated version using underscorejs   function foo l  target        var off   target -   reduce l  function acc  x    return acc   Math round x     0       return   chain l               sortBy function x    return Math round x  - x                 map function x  i    return Math round x     off  gt  i  -  i  gt    l length   off                   value       foo  13 626332  47 989636  9 596008  28 788024   100       gt   48  29  14  9  foo  16 666  16 666  16 666  16 666  16 666  16 666   100       gt   17  17  17  17  16  16  foo  33 333  33 333  33 333   100       gt   34  33  33  foo  33 3  33 3  33 3  0 1   100       gt   34  33  33  0

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check if this is valid or not as far as my test cases I am able to get this working   let s say number is k    sort percentage by descending oder  iterate over each percentage from descending order  calculate percentage of k for first percentage take Math Ceil of output  next k   k-1 iterate over till all percentage is consumed

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You could try keeping track of your error due to rounding  and then rounding against the grain if the accumulated error is greater than the fractional portion of the current number   13 62 - gt  14    38  47 98 - gt  48    02    40 total    9 59 - gt  10    41    81 total   28 78 - gt  28  round down because  81  gt   78  ------------         100   Not sure if this would work in general  but it seems to work similar if the order is reversed   28 78 - gt  29    22   9 59 - gt   9  - 37  rounded down because  59  gt   22  47 98 - gt  48  - 35  13 62 - gt  14    03  ------------         100   I m sure there are edge cases where this might break down  but any approach is going to be at least somewhat arbitrary since you re basically modifying your input data

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If you really must round them  there are already very good suggestions here  largest remainder  least relative error  and so on    There is also already one good reason not to round  you ll get at least one number that  looks better  but is  wrong    and how to solve that  warn your readers  and that is what I do   Let me add on the  wrong  number part   Suppose you have three events entitys     with some percentages that you approximate as   DAY 1 who    real   app ---- ------- ------   A   33 34    34   B   33 33    33   C   33 33    33   Later on the values change slightly  to  DAY 2 who    real   app ---- ------- ------   A   33 35    33   B   33 36    34   C   33 29    33   The first table has the already mentioned problem of having a  wrong  number  33 34 is closer to 33 than to 34   But now you have a bigger error  Comparing day 2 to day 1  the real percentage value for A increased  by 0 01   but the approximation shows a decrease by 1    That is a qualitative error  probably quite worse that the initial quantitative error   One could devise a approximation for the whole set but  you may have to publish data on day one  thus you ll not know about day two  So  unless you really  really  must approximate  you probably better not

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Note  the selected answer is changing the array order which is not preferred  here I provide more different variations that achieving the same result and keeping the array in order Discussion given  98 88   56   56  how do you want to round it  you have four option 1- round things up and subtract what is added from the rest of the numbers  so the result becomes  98  1  1  this could be a good answer  but what if we have  97 5   5   5   5   5   5   then you need to round it up to  95  1  1  1  1  1  do you see how it goes  if you add more 0-like numbers  you will lose more value from the rest of your numbers  this could be very troublesome when you have a big array of zero-like number like  40   5   5          5   when you round up this  you could end up with an array of ones   1  1         1  so round-up isn t a good option  2- you round down the numbers  so  98 88   56   56  becomes  98  0  0   then you are 2 less than 100  you ignore anything that is already 0  then add up the difference to the biggest numbers  so bigger numbers will get more  3- same as previous  round down numbers  but you sort descending based on the decimals  divide up the diff based on the decimal  so biggest decimal will get the diff  4- you round up  but you add what you added to the next number  so like a wave what you have added will be redirected to the end of your array  so  98 88   56   56  becomes  99  0  1  none of these are ideal  so be mindful that your data is going to lose its shape  here I provide a code for cases 2 and 3  as case No 1 is not practical when you have a lot of zero-like numbers   it s modern Js and doesn t need any library to use 2nd case const v1    13 626332  47 989636  9 596008  28 788024       gt    14  48  9  29   const v2    16 666  16 666  16 666  16 666  16 666  16 666       gt    17  17  17  17  16  16    const v3    33 333  33 333  33 333       gt    34  33  33   const v4    33 3  33 3  33 3  0 1       gt    34  33  33  0   const v5    98 88   56   56       gt   100  0  0   const v6    97 5   5   5   5   5   5       gt    100  0  0  0  0  0    const normalizePercentageByNumber    input    gt        const rounded  number     input map x   gt  Math floor x        const afterRoundSum   rounded reduce  pre  curr    gt  pre   curr  0       const countMutableItems   rounded filter x   gt  x  gt  1  length      const errorRate   100 - afterRoundSum           const deductPortion   Math ceil errorRate   countMutableItems            const biggest       rounded  sort  a  b    gt  b - a  slice 0  Math min Math abs errorRate   countMutableItems        const result   rounded map x   gt            const indexOfX   biggest indexOf x           if  indexOfX  gt   0                x    deductPortion              console log biggest              biggest splice indexOfX  1               return x                    return x              return result     3rd case const normalizePercentageByDecimal    input  number      gt         const rounded  input map  x  i    gt    number  Math floor x   decimal  x 1  index  i           const decimalSorted      rounded  sort  a b   gt  b decimal-a decimal            const sum   rounded reduce  pre  curr   gt  pre   curr number  0        const error  100-sum           for  let i   0  i  lt  error  i              const element   decimalSorted i           element number               const result      decimalSorted  sort  a b   gt  a index-b index            return result map x  gt  x number       4th case you just need to calculate how much extra air added or deducted to your numbers on each roundup and  add or subtract it again in the next item  const v1    13 626332  47 989636  9 596008  28 788024       gt   14  48  10  28   const v2    16 666  16 666  16 666  16 666  16 666  16 666       gt   17  16  17  16  17  17  const v3    33 333  33 333  33 333       gt   33  34  33  const v4    33 3  33 3  33 3  0 1       gt   33  34  33  0   const normalizePercentageByWave  v4 reduce  pre  curr  i  arr    gt         let number   Math round curr   pre decimal       let total   pre total   number       const decimal   curr - number       if  i    arr length - 1  amp  amp  total  lt  100            const diff   100 - total          total    diff          number    diff             return   total  numbers      pre numbers  number   decimal          total  0  numbers      decimal  0

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I wrote a C  version rounding helper  the algorithm is same as Varun Vohra s answer  hope it helps     public static List lt decimal gt  GetPerfectRounding List lt decimal gt  original      decimal forceSum  int decimals        var rounded   original Select x   gt  Math Round x  decimals   ToList        Debug Assert Math Round forceSum  decimals     forceSum       var delta   forceSum - rounded Sum        if  delta    0  return rounded      var deltaUnit   Convert ToDecimal Math Pow 0 1  decimals     Math Sign delta        List lt int gt  applyDeltaSequence       if  delta  lt  0                applyDeltaSequence   original              Zip Enumerable Range 0  int MaxValue    x  index    gt  new   x  index                 OrderBy a   gt  original a index  - rounded a index                ThenByDescending a   gt  a index               Select a   gt  a index  ToList              else               applyDeltaSequence   original              Zip Enumerable Range 0  int MaxValue    x  index    gt  new   x  index                 OrderByDescending a   gt  original a index  - rounded a index                Select a   gt  a index  ToList               Enumerable Repeat applyDeltaSequence  int MaxValue           SelectMany x   gt  x           Take Convert ToInt32 delta deltaUnit            ForEach index   gt  rounded index     deltaUnit        return rounded      It pass the following Unit test    TestMethod  public void TestPerfectRounding         CollectionAssert AreEqual Utils GetPerfectRounding          new List lt decimal gt   3 333m  3 334m  3 333m   10  2           new List lt decimal gt   3 33m  3 34m  3 33m         CollectionAssert AreEqual Utils GetPerfectRounding          new List lt decimal gt   3 33m  3 34m  3 33m   10  1           new List lt decimal gt   3 3m  3 4m  3 3m         CollectionAssert AreEqual Utils GetPerfectRounding          new List lt decimal gt   3 333m  3 334m  3 333m   10  1           new List lt decimal gt   3 3m  3 4m  3 3m          CollectionAssert AreEqual Utils GetPerfectRounding          new List lt decimal gt    13 626332m  47 989636m  9 596008m  28 788024m    100  0           new List lt decimal gt   14  48  9  29        CollectionAssert AreEqual Utils GetPerfectRounding          new List lt decimal gt    16 666m  16 666m  16 666m  16 666m  16 666m  16 666m    100  0           new List lt decimal gt    17  17  17  17  16  16         CollectionAssert AreEqual Utils GetPerfectRounding          new List lt decimal gt    33 333m  33 333m  33 333m    100  0           new List lt decimal gt    34  33  33         CollectionAssert AreEqual Utils GetPerfectRounding          new List lt decimal gt    33 3m  33 3m  33 3m  0 1m    100  0           new List lt decimal gt    34  33  33  0

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I m not sure what level of accuracy you need  but what I would do is simply add 1 the first n numbers  n being the ceil of the total sum of decimals  In this case that is 3  so I would add 1 to the first 3 items and floor the rest  Of course this is not super accurate  some numbers might be rounded up or down when it shouldn t but it works okay and will always result in 100    So   13 626332  47 989636  9 596008  28 788024   would be  14  48  10  28  because Math ceil  626332  989636  596008  788024     3  function evenRound  arr       var decimal   - arr map function  a    return a   1         reduce function  a b    return a   b        Ceil of total sum of decimals   for   var i   0  i  lt  decimal    i         arr  i       arr  i       compensate error by adding 1 the the first n items       return arr map function  a    return   a        floor all other numbers    var nums   evenRound    13 626332  47 989636  9 596008  28 788024      var total   nums reduce function  a b    return a   b         gt  100   You can always inform users that the numbers are rounded and may not be super-accurate

User · Answer

For those having the percentages in a pandas Series  here is my implemantation of the Largest remainder method  as in Varun Vohra s answer   where you can even select the decimals to which you want to round   import numpy as np  def largestRemainderMethod pd series  decimals 1        floor series     10  decimals   pd series  astype np int   apply np floor      diff   100    10  decimals  - floor series sum   astype np int      series decimals   pd series - floor series    10  decimals      series sorted by decimals   series decimals sort values ascending False       for i in range 0  len series sorted by decimals            if i  lt  diff              series sorted by decimals iloc  i     1         else              series sorted by decimals iloc  i     0      out series     floor series   series sorted by decimals     10  decimals   sort values ascending False       return out series

[algorithm] How to make rounded percentages add up to 100%

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