Question

What do you mean by the propagation of errors? Explain the propagation of errors in addition and multiplication.

Answer 1

A number of measured quantities may be involved in the final calculation of an experiment. Different types of instruments might have been used for taking readings. Then we may have to look at the errors in measuring various quantities, collectively.

The error in the final result depends on

(i) The errors in the individual measurements

(ii) On the nature of mathematical operations performed to get the final result. So we should know the rules to combine the errors.
The various possibilities of the propagation or combination of errors in different mathematical operations are discussed below:

(i) Error in the sum of two quantities
Let ∆A and ∆B be the absolute errors in the two quantities A and B respectively. Then, Measured value of A = A ± ∆A

The measured value of B = B ± ∆B

Consider the sum, Z = A + B

The error ∆Z in Z is then given by

Z ± ∆Z = (A ± ∆A) + (B ± ∆B)

= (A + B) ± (∆A + ∆B) = Z ± (∆A + ∆B)

(Or) ∆Z = ∆A + ∆B

The maximum possible error in the sum of two quantities is equal to the sum of the absolute errors in the individual quantities.
Error in the product of two quantities: Let ∆A and ∆B be the absolute errors in the two quantities A, and B, respectively. Consider the product Z = AB

The error ∆Z in Z is given by Z ± ∆Z = (A ± ∆A) (B ± ∆B)

= (AB) ± (A ∆ B) ± (B ∆ A) ± (∆A • ∆B)

Dividing L.H.S by Z and R.H.S by AB, we get,

1 ± `"∆Z"/"Z" = 1 ± "∆B"/"B" ± "∆A"/"A" ± "∆A"/"A" . "∆B"/"B"`

As ∆A/A, ∆B/B are both small quantities, their product term `"∆A"/"A" . "∆B"/"B"` can be neglected.

The maximum fractional error in Z is `"∆Z"/"Z" = ± ("∆A"/"A" + "∆B"/"B")`