Generally, the result of an experiment is obtained by doing mathematical operations on several measurements. Obviously, the final error depends not only upon the errors in individual measurements but also on the nature of mathematical operations.
Following are the rules for combination of errors.
(a) Errors in a Sum or Difference: Let there be two quantities A and B. If ΔA and ΔB are the corresponding absolute errors in their measurements, then
Measured value of A = A ± ΔA
Measured value of B = B ± ΔB
(i) Let Z denote the sum of A and B and ΔZ be the corresponding absolute error.
Clearly, Z = A + B and
Z ± ΔZ (A + ΔA) + (B + ΔB)
Z ± ΔZ (A + B) + (ΔA ± ΔB)
ΔZ = ± (ΔA ± ΔB) … (1)
Thus, the maximum error in Z, i.e., ΔZ = (ΔA ± ΔB), i.e., the maximum error in the sum is the sum of individual errors.
(ii) In case Z denotes the difference of A and B, and ΔZ is the corresponding absolute error,
Z = A – B
Clearly, Z ± ΔZ (A ± ΔA) – (B ± ΔB)
= A ± ΔA – B ± ΔB
Or ± ΔZ = ± (ΔA + ΔB) … (2)
Thus, the maximum error in Z, i.e., ΔZ = (ΔA ± ΔB) the maximum error in the difference is again the sum of the individual errors.
Hence, when two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors associated with the measurements of the quantities to be added or subtracted.
(b) Errors in a Product or Quotient: Let there be two quantities A and B and let AA and AB be the corresponding absolute errors in their measurements. Clearly,
Measured value of A = A ± ΔA
Measured value of B = B ± ΔB
(i) Let Z denote the product of A and B and let ΔZ be the corresponding absolute error.
Clearly, Z = AB and
Z + ΔZ = (A ± ΔA) (B ± ΔB)
Or Z + ΔZ = AB ± BΔA ± AΔB ± ΔAΔB)
Dividing both sides by Z (= AB), we get
Or 1 ± ΔZ/Z = 1 ΔA/A ± ΔB/B
\(1\pm \frac{\Delta Z}{Z}=\frac{AB}{AB}\pm \frac{B\Delta A}{AB}\pm \frac{A\Delta B}{AB}\pm \frac{\Delta A\Delta B}{AB}\).
(We have ignored ΔAΔB/AB as it contains the product of two small quantities ΔA and ΔB)
Thus, ±ΔZ/Z = ± (ΔA/A + ΔB/B)
Or ΔZ/Z = ΔA/A + ΔB/B … (3)
ΔZ/Z, ΔA/A, ΔB/B are the relative errors in the measurement of Z, A and B respectively.
(ii) Let Z denote the quotient of A and B and let ΔZ be the corresponding absolute error.
Clearly, Z = A/B
\(Z\pm \Delta Z=\frac{\left( A\pm \Delta A \right)}{\left( B\pm \Delta B \right)}=\frac{A\left( 1\pm \frac{\Delta A}{A} \right)}{B\left( 1\pm \frac{\Delta B}{B} \right)}\).
\(=Z\left( 1\pm \frac{\Delta A}{A} \right){{\left( 1\pm \frac{\Delta B}{B} \right)}^{-1}}\).
\(=Z\left( 1\pm \frac{\Delta A}{A} \right)\left( 1\pm \frac{\Delta B}{B} \right)\,\) (Applying binomial theorem).
\(=Z\left[ 1\pm \frac{\Delta A}{A}\pm \frac{\Delta B}{B}+\frac{\Delta A\Delta B}{AB} \right]\).
[Neglecting the last term s it is the product of two small quantities ΔA and ΔB]
Or ±ΔZ = ±Z ΔA/A ± ΔB/B
Or ±ΔZ/Z = ±ΔA/A ± ΔB/B
Or ΔZ/Z = ΔA/A + ΔB/B … (4)
We find that the results (3) and (4) are the same.
Thus, when two quantities are multiplied (or divided), the relative error in the product (or quotient, i.e., division) is the sum of the relative errors in the quantities to be multiplied or divided.
(c) Error Due to the Power of a Measured Quantity: Let there be a quantity A and let ΔA be the absolute errors in its measurement.
Measured values of A = A ± ΔA
Let Z = A²
Clearly, Z ± ΔZ = (A ± ΔA)²
= A² + ΔA² ± 2A ΔA
Neglecting (ΔA)² as it is exceedingly small,
Z ± ΔZ = A² + ΔA² ± 2A ΔA
Or ± ΔZ = ± 2A ΔA
Or ΔZ = 2A ΔA
Dividing both sides by Z, we get
ΔZ/ Z = 2A ΔA/ Z
= 2A ΔA/ A² = 2 ΔA/ A
Thus, the relative error in Z (i.e. A2) is twice the relative error in A.
(d) General Relation for Error involving Powers of Measured Quantities:
If, \(Z=\frac{{{A}^{p}}{{B}^{q}}}{{{C}^{r}}}\), then
\(\frac{\Delta Z}{Z}=p\frac{\Delta A}{A}+q\frac{\Delta B}{B}+r\frac{\Delta C}{C}\)… (5)