|
When you want to multiply or divide 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 fractional standard
deviations, add them, then take the square root of the sum to get the fractional
total deviation. If you have values A +- dA, B +- dB, ... and want to compute X = A*B*..., the total error
dX is then
- dX/X = sqrt( (dA/A)2 + (dB/B)2 +...)
Note that you will add the
squares of the errors even if you are dividing 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
problem is trivial:
- (1.67 * 5.23)/1.88 = 4.65
To compute the standard deviation of the
result, we must sum the squares of the relative errors and then take the sqaure
root.
- dX/4.65 = sqrt( (0.05/1.67)2 +
(0.09/5.23)2 + (0.07/1.88)2)
- Stot/4.65 = sqrt(0.000896 + 0.000296 + 0.00139)
- Stot/4.65 = 0.0508
- Stot = 0.236
Since we only report error to 1 significant
figure, the answer to this problem would be 4.7+-0.2
See also the page on handling addition and
subtraction 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 X = 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.
Example: You want to perform the calculation X = 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) = 0.417
The relative error is the square root of the sum of squares of the relative
errors
- dX/0.417 = sqrt( (0.01/14.99)2 +
(0.09/3.56)2 + (0.5/10.1)2 + (0.006/1)2)
- dX/0.417 = sqrt(4.45*10-7 +
6.39*10-4 + 2.45*10-3 + 3.60*10-5)
- dX/0.417 = sqrt(3.13*10-3)
- dX/0.417 = 0.0559
- dX = 0.0233
The answer would thus be
reported as 0.42 +- 0.02. Note that the vast majority of the error came from one
piece of data: 10.1 +- 0.5. It's often the case in experiments that one set of
data has a much larger error than the rest. This is known as the limiting
error- you may know the temperature and pressure to 8 significant figures,
but if you only have the volume to 2 the error in the volume will be the
limiting error. |