ERROR PROPAGATION RULES
FOR ELEMENTARY OPERATIONS AND FUNCTIONS

Let R be the result of a calculation, without consideration of errors, and ΔR be the error (uncertainty) in that result. Determinate errors have determinable sign and constant size. Indeterminate errors have unpredictable size and sign, with equal likelihood of being + or -.

RULES FOR ELEMENTARY OPERATIONS (DETERMINATE ERRORS)

SUM RULE:          When R = A + B        then    ΔR = ΔA + ΔB 

DIFFERENCE RULE:   When R = A - B        then    ΔR = ΔA - ΔB 

PRODUCT RULE:      When R = AB           then    (ΔR)/R = (ΔA)/A + (ΔB)/B 

QUOTIENT RULE:     When R = A/B          then    (ΔR)/R = (ΔA)/A + (ΔB)/B 

POWER RULE:        When R = An           then     (ΔR)/R = n(ΔA)/A  

                                         or       (ΔR) = n An-1(ΔA)

Memory clues:

When quantities are added (or subtracted) their absolute errors add (or subtract). But when quantities are multiplied (or divided), their relative fractional errors add (or subtract).

These rules will be freely used, when appropriate.

We can also collect and tabulate the results for commonly used elementary functions. Note: Where Δt appears, it must be expressed in radians.

RULES FOR ELEMENTARY FUNCTIONS (DETERMINATE ERRORS)

EQUATION              ERROR EQUATION 

R = sin q             ΔR = (dq) cos q 

R = cos q             ΔR = -(dq) sin q 

R = tan q             ΔR = (dq) sec2 q 

R = ex                ΔR = (Δx) ex 

R = e-x               ΔR = -(Δx) e-x 

R = ln(x)             ΔR = (Δx)/x

Any measures of error may be converted to relative (fractional) form by using the definition of relative error. The fractional error in x is: fx = (ΔR)x)/x where (ΔR)x is the absolute ereror in x. Therefore xfx = (ΔR)x.

The rules for indeterminate errors are simpler.

RULES FOR ELEMENTARY OPERATIONS (INDETERMINATE ERRORS)

SUM OR DIFFERENCE:          When R = A + B        then    ΔR = ΔA + ΔB 

PRODUCT OR QUOTIENT:        When R = AB           then    (ΔR)/R = (ΔA)/A + (ΔB)/B 

POWER RULE:                 When R = An           then     (ΔR)/R = n(ΔA)/A  

                                                  or       (ΔR) = n An-1(ΔA)

The indeterminate error rules for elementary functions are the same as those for determinate errors except that the error terms on the right are all positive.

Students who are taking calculus will notice that these rules are entirely unnecessary. The determinate error equations may be found by differentiating R, then replading dR, dx, dy, etc. with ΔR, Δx, Δy, etc. This is equivalent to expanding ΔR as a Taylor series, then neglecting all terms of higher order than 1. This is a valid approximation when (ΔR)/R, (Δx)/x, etc. are all small fractions.
The indeterminate error equations may be constructed from the determinate error equations by algebraically reaarranging the final resultl into standard form:

ΔR = ( )Δx + ( )Δy + ( )Δz .
The coefficients in parantheses ( ), and/or the errors themselves, may be negative, so some of the terms may be negative. Now make all negative terms positive, and the resulting equuation is the correct indeterminate error equation.

Click here for a printable summary sheet Strategies of Error Analysis.