Question

Derive the formula for propagation of errors in following case :- sum of two quantities.

Answer 1

In this article, we shall study the propagation of errors in different mathematical operations.

Suppose a result x is obtained by addition of two quantities say a and b

i.e. x = a + b

Let Δ a and Δ b are absolute errors in the measurement of a and b and Δ x be the corresponding absolute error in x.

∴ x ± Δ x = ( a ± Δ a) + ( b ± Δ b)

∴ x ± Δ x = ( a + b ) ± ( Δ a + Δ b)

∴ x ± Δ x = x ± ( Δ a + Δ b)

∴ ± Δ x = ± ( Δ a + Δ b)

∴ Δ x = Δ a + Δ b

Thus maximum absolute error in x = maximum absolute error in a + maximum absolute error in b

Thus, when a result involves the sum of two observed quantities, the absolute error in the result is equal to the sum of the absolute error in the observed quantities.