Combination of Errors

• a) Error of a sum or a difference

Suppose Z = A + B. --------------(i)

Let ΔA = absolute error in measurement of quantity A
ΔB = absolute error in measurement of quantity B
 ΔZ = absolute error in sum Z of A and B.
Then A ± ΔA = measured value of A
B ± ΔB = measured value of B.
Z ± Δ Z = measured value of sum Z of A and B.
So, (i) becomes
Z ± Δ Z = (A ± ΔA) + (B ± ΔB)
Z ± Δ Z = (A+ B ) ± (ΔA+ ΔB)
Or Δ Z = ± ΔA± ΔB
So possible errors in Z are +ΔA+ ΔB, -ΔA+ ΔB, +ΔA- ΔB, -ΔA- ΔB.
& maximum possible error in Z = ± (ΔA+ ΔB)
Similarly, for the difference Z = A – B, we have
Z ± Δ Z = (A ± ΔA) – (B ± ΔB)
= (A – B) ± ΔA ± ΔB
or, ± ΔZ = ± ΔA ± ΔB
The maximum value of the error ΔZ is again Z = ± (ΔA+ ΔB)
Hence the rule : When two quantities are added or subtracted, the absolute error in
the final result is the sum of the absolute errors in the individual quantities.

• (b) Error of a product or a quotient

Suppose Z = A B. -------------- (i)
Let ΔA = absolute error in measurement of quantity A
ΔB = absolute error in measurement of quantity B
 ΔZ = absolute error in product Z of A and B.
Then A ± ΔA = measured value of A
B ± ΔB = measured value of B.
Z ± Δ Z = measured value of product Z of A and B.
Z ±ΔZ = (A ± ΔA) (B ± ΔB)

             = AB ± B ΔA ± A ΔB ± ΔA ΔB.
Dividing LHS by Z and RHS by AB we have,

Since ΔA and ΔB are small, we shall ignore their product.
Hence the maximum relative error
ΔZ/ Z = (ΔA/A) + (ΔB/B).
Similarly, for division we also have ΔZ/ Z = (ΔA/A) + (ΔB/B
Hence the rule: When two quantities are multiplied or divided, the relative error in
the result is the sum of the relative errors in the multipliers.