## Following Errors in Computations

## Following Errors in Computations

(OP)

I hope that I am not off base for this forum. Long long time ago at Cornell, the physics department had a paper that showed how errors propagate through compuatations. Example, if I measure the L and B of a table and use a ruler to the nearest 1 inch, my measurement error is 0.5". Hence my error is LxB + - "??". If my measurement was to mm, then my error is 0.5mm and the accuracy of my final answer is better.

Does anyone have those relationships for adding, subtracting, multiplying, dividing, log, power raising?????

Does anyone have those relationships for adding, subtracting, multiplying, dividing, log, power raising?????

## RE: Following Errors in Computations

(... 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) = n

^{2}*V(A)V(A*B) = B

^{2}*V(A) + A^{2}*V(B)V(A/B) = {B

^{2}*V(A) + A^{2}*V(B)} / B^4V(A

^{n}) = {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.

## RE: Following Errors in Computations

http://mathworld.wolfram.com/ErrorPropagation.html

TTFN

## RE: Following Errors in Computations

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

http://www.cs.utep.edu/interval-comp/

http://www.mat.univie.ac.at/~neum/interval.html

## RE: Following Errors in Computations

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.

## RE: Following Errors in Computations

## RE: Following Errors in Computations

## RE: Following Errors in Computations

## RE: Following Errors in Computations

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.