Units and Dimensions - Combination of Errors

Description:

When measured quantities with errors associated with them are added, subtracted, multiplied or divided, their errors can also combine in various ways.

Addition and Subtraction

Consider two measured quantities

a = a₀ ± Δa

b = b₀ ± Δb

When they are added, or subtracted, their absolute errors always add up.

Therefore, if R = a ± b and R = R₀ + ΔR, then −

R = a₀ ± b₀

ΔR = Δa + Δb

Multiplication and Division

Consider two measured quantities −

a = a₀ ± Δa

b = b₀ ± Δb

When they are multiplied, or divided, their relative errors always add up

Therefore, if R = a × b or R = a/b; and R = R₀ + ΔR, then −

R₀ = a₀ × b₀ or R₀ = a₀/b₀

ΔR/R = Δa/a₀ + Δb/b₀

Raised to an Exponent

Consider a measured quantity −

a = a₀ ± Δa

When it is raised to a power of n, its relative error is multiplied by n.

Therefore, if R = aⁿ, R = R₀ + ΔR, then −

R₀ = a₀ⁿ

ΔR/R = n (Δa/a₀)