Following Errors in Computations 7

I had to work these out for myself many years ago. The results are below, but since I have never found them in any texts I can not guarantee that they are correct. However, I was slightly encouraged by their vague resemblances to some basic results in the differential calculus.

(... hope I can drive the TGML properly here ...)

Let A be the best estimate of a quantity;
Let B be the best estimate of another quantity which is statistically independent of A;
Let V(A) and V(B) be the respective variances associated with the estimates of A and B;
Let n be a numerical constant (exactly known).

Then the following formulae will assist in determining the variance associated with the results of arithmetic operations involving A, B and n.

V(A+B) = V(A) + V(B)
V(A-B) = V(A) + V(B)
V(n*A) = n²*V(A)
V(A*B) = B²*V(A) + A²*V(B)
V(A/B) = {B²*V(A) + A²*V(B)} / B^4
V(Aⁿ) = {nA^n-1}² * V(A)
V(A^B) = {BA^B-1}² * V(A) + {A^B*ln(A)}²*V(B)

The first three formulae are exact, and can be found in any basic text book on statistics. The remaining formulae are approximate, and are accurate only when sqrt(V(A))/A and sqrt(V(B))/B are small. In most engineering situations this condition is adequately satisfied.

Note that these formulae are valid only for true variances in the strict statistical sense of that term (ie the square of the standard deviation).

Note also the requirement for statistical independence.

If anyone disagrees with these, please advise via this forum.

The traditional propagation of errors analysis is done with partial derivatives, see:


TTFN

There is a mathematical disciplin called Interval Analysis that is devoted to studying methods for calculating with uncertain quantities, where the uncertainty is specified as intervals (e.g. a known measurement error on a quantity, in your example).

The rules for calculation is defined by the interval arimethic. For more information check out the following sites:

A good article for propagation of error is in NBS Special Publication 300, Vol. 1 (1969), called "Precision Measurement and Calibration". The article is by Harry Ku (also the editor of the volume); the article is entitled "Notes on the Use of Propagation of Error Formulas". It's a bit more accessible discussion of Cramer's theorem, which is the basis of the statistical treatment (see any text on mathematical statistics).

Also in that NBS publication is a good article entitled "Computations with Approximte Numbers" by Delury. It's a grass-roots discussion of interval arithmetic.

You also want to be careful to make sure your stochastic variables meet with the assumptions made by these techniques. In the 80's I remember working with data that wasn't amenable to these techniques and I had to resort to Monte Carlo modelling to get quantitative answers.

For quick back of the envelope calculations, a good technique is the thing we were all probably taught in a basic physics or math class: you can approximate the uncertainty of a dependent variable by using differentials. For example, if z = f(x, y), then dz = f_x*dx + f_y*dy where f_x means the partial derivate with respect to x. Even if you can't do the calculation easily in analytical form, it's not hard to approximate things with numbers and a calculator.

I ran across a good "lecture" while surfing one day - sorry I didn't write down the URL but details are: NAU.edu Course CHM320 - in case anyone wants to try to find it himself. If I could get a coordinate to send, I'd pass it on.

I know this question has been around for a while, but there is a publication regarding how to handle errors. I came across it in a physics class and have found it to be useful. The title is An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. It was written by John R. Taylor.

DGeorgeson - was this from Cornell? That is where my copy (can't remember author) is from - and, as I don't have it handy, is why I asked the question!

John Taylor is/was at the Univ of Colorado, Boulder, Dept of physics. Anyway--

A = first value
a = error of A
B = second value
b = error of b
C = result of A and B
c = propagated error of a and b
D = known constant w/o error
Example

Add
A + B = C
c = (a^2 + b^2)^0.5
Subtract
A - B = C
c = (a^2 + b^2)^0.5
Multiply
A * B = C
c/C = (a/A) + (b/B)
Divide
A / B = C
c/C = (a/A) + (b/B)
Power
C = A^D
c/C = D*(a/A)

These assume that the values and uncertainties are independent.