The result of an experiment is generally calculated by using some formulae which may contain many measured quantities and only one unknown quantity operations (e.g. addition, subtraction, multiplication, division etc.) may be carried out. Therefore errors in observations propagate to result.
(i) Error due to addition of quantities: Suppose two quantities A and B are related with third quantity Z as under:
Z = (A+B) .........(1)
Errors in measurement of A&B are ± ∆A and ± ∆B respectively. Error propagated to Z is ± ∆Z.
Therefore equation (1) can be written as,
(Z ± ∆Z) = (A ± ∆A) ÷ (B ± ∆B)
or Z ± ∆Z = (A + B) ± (∆A + ∆B)
or Z ± ∆Z = Z± (∆A + ∆B)
or ± ∆Z = ± (∆A + ∆B)
∴ Maximum absolute error,
\(|\bigtriangleup Z|_{max} = (\bigtriangleup A + \bigtriangleup B)\)
(ii) Error in difference of quantities: Suppose the difference of two quantities A and B is related with third quantity Z as under:
Z = (A - B)
Suppose errors in measurement of A and B are ∆A and ∆B respectively. The error propagated to Z is ∆Z.
∴ (Z ± ∆Z) = (A ± ∆A) - (B ± ∆B)
or Z ± ∆Z = (A - B) ± (∆A + ∆B)
or Z ± ∆Z = Z ± (∆A + ∆B)
or ± ∆Z = ±(∆A + ∆B)
∴|∆Z|max = ∆A + ∆B
(iii) Error in product of quantities
Suppose two quantities A and B are related with quantity Z as under.
Z = A x B .........(1)
If absolute errors in A, B and Z are ∆A, ∆B and ∆Z respectively, then equation (1) will become now:
∵ \(\frac{\bigtriangleup A}{A} \times \frac{\bigtriangleup B}{B}\) is very small quantity, therefore it can be neglected.
This is maximum relative error propagated to Z.
(iv) Error in quotientt of quantities
(v) Error due to measured quantity raised to power
Suppose Z = A2 = A x A
∴ log Z = log A + log A
On differentiating both sides,
In general: If Z = Am
The log Z = log Am = m log A
On differentating both sides,
(vi) Maximum relative error in mixed quantities
