Themes > Science > Chemistry > Miscellenous > Help file Index > Error Analysis > Addition and Subtraction of Values with Standard Deviation
 

When you want to add or subtract average values reported with their standard deviations, you don't simply add the standard deviations to produce the final standard deviation. Instead, you square the standard deviations, add them, then take the square root of the sum.

Stot = sqrt(S12 + S22 + ...)
Note that you will add the squares of the errors even if you are subtracting the actual values.

An example should make this clearer. Assume we have the following three values with their standard deviations

  • A = 1.67 +- 0.05
  • B = 5.23 +- 0.09
  • C= 1.88 +- 0.07
and we want to compute X = A+B-C. The actual addition of the values themselves is trivial:
1.67 + 5.23 - 1.88 = 5.02
To compute the standard deviation of the result, we must sum the squares of the errors and then take the sqaure root.
Stot = sqrt(0.052 + 0.092 + 0.072)
Stot = sqrt(0.0025+0.0081+0.0049)
Stot = 0.12
Note that we added 0.072, the error of 1.88, even though we subtracted 1.88 from the result. Since we typically only report standard deviations to 1 significant figure, we would report the result of this sum as
5.0+-0.1

The reason we do not simply add the errors is that this overstates the probable error. If we just added the errors, we would be assuming that all the errors were maximum and all in the same direction. This isn't likely however- chances are that an error in one set of data may help cancel an error in another set.

See also the page on handling multiplication and division with errors. If you are given a problem that has both addition/subtraction and multiplication/division, you should work through the errors in the same order that you do the math: multiplication/division before addition/subtraction. For example, if you had to compute A*B + C, work out the error in A*B by the multiplication rule, then work out the error in (A*B) + C by the addition rule.

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Example: You want to perform the calculation A+B-C-D. What is the result and the standard deviation of the result?

  • A = 14.99 +- 0.01
  • B = 3.56 +- 0.09
  • C = 10.1 +- 0.5
  • D = 1.000 +- 0.006

Solution: The average result is easy to compute:

14.99 + 3.56 - 10.1 -1.000 = 7.5
(Note the loss in significant figures when adding. This will be borne out by the error computed below.)

The error is the square root of the sum of squares

Stot = sqrt(0.012 + 0.092 + .52 + 0.0062)
Stot = sqrt(0.258236)
Stot = 0.508
Since we report error to 1 SF, this is +- 0.5, and the final result is 7.5 +- 0.5. Note that removing the sig figs during the addition was correct.


Information provided by: http://learn.chem.vt.edu