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# Sum of all weights equal to 1

Posted on 2011-03-22
Medium Priority
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Last Modified: 2012-06-21
I have the following formula: can i add something to it to be sure that the sum of all the weight is equal to 1?
the weights are between 0.1 and 0.9:
Where N = 11 number of elements to be weighted
i = index of each element
Min = 0.1
Max = 0.9

value = min +(max - min)* (i-1.0)/(N-1)

That's shouldn't be so hard for experts no?
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Question by:dadadude
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8 Comments

Author Comment

ID: 35190731
Found the solution:
When you use your own weight set in an interpolation procedure, you will have the option of normalizing the weights.  This means that each individual weight value (geographic multiplied by user-assigned weight) is divided by the sum of the weights assigned to all of the units being interpolated. Thus, the sum of all weights will equal one.  This option allows you to compute a population-weighted average when applied to rate data weighted by a population dataset, or by other types of count data.
http://www.biomedware.com/files/documentation/spacestat/Statistics/Interpolation/Spatial_Interpolation.htm#Normalize

Thank you. Hope that it can help other also.
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LVL 31

Expert Comment

ID: 35195096
The principle is as follows:-

Given n values

W1, W2, W3.....Wn

then define their sum S as

S = W1 + W2 + W3.....+ Wn

then the values

W1/S, W2/S, W3/S.....Wn/S

will add up to 1.

As

W1/S + W2/S + W3/S.....+ Wn/S =  ( W1 + W2 + W3.....+ Wn )/S = S/S = 1

0

LVL 37

Expert Comment

ID: 35201149
That will normalize the values for the particular set (which may be what you want) but if you want the normalization to be standardized on different sets then you should divide by max*N. That way, any set of the same size is normalized the same way so you won't give preference to some values that shouldn't get it.
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LVL 3

Expert Comment

ID: 35204879
I was almost tricked into solving an impossible equation.  ha ha ha...  This is a trick question.

If you have N = 11 and the weight must be a minimum of 0.1, there is no way that that the sum can ever be 1.  The minimum sum of the weights would be 1.1 and no possible solution.

If N = 10, there is only one possible solution for the weights since the weight must be a minimum of 0.1.  All the weights must be 0.1.

Therefore, N needs to be less than 10 in order to have a logical equation, if the minimum is 0.1.  LOL  Good one.
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Author Comment

ID: 35208351
0.1      0.066666667
0.2      0.133333333
0.3      0.2
0.4      0.266666667
0.5      0.333333333

1.5      1

1.5 is the summation of 0.1 + 0.2 + 0.3 + 0.4 + 0 .5 = 1.5
u devide all the other by 1.5 and add them u'll get 1 lol
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Author Comment

ID: 35208360
well of course the sum will never be 1 if they are not normalized that's for sure
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LVL 37

Expert Comment

ID: 35209622
If you want the weight to always sum to exactly 1 then you just divide each one by the sum of all of them as you mentioned.
If you want the weights to be normalized in a more standardized way, you should divide by max*N as I previously mentioned.
If you want suggestions for which to use, what application is this for?
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LVL 3

Accepted Solution

InfoStranger earned 2000 total points
ID: 35292244
If you normalize the data as you said, your weights are the normalized data not the weights.  Which means that your weights are really meaningless.

They are just arbitrary figures.

All that means is that you randomly pick any 11 numbers sum them up and divide each number by the sum.  It will always add up to 1.  Weights are meaningless in the question.

Like what GwynforWeb said.  You new weights are Wi/S lets call that Zi.

So, now your new weights are Zi the normalized weights.  All you really did was shift all your data to fit.  Your min and max are not needed as well, since you have a new weight.  You will need to shift the min and max on the new weight to be consistent.

The question does not make sense unless you remove the minimum from the weight.
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