Propagation of Error Product

Description:

When two measurements with error are multiplied then their product will have fractional error equal to sum of fractional errors of the numbers.

Consider two measurement – a with error Δa and b with error Δb that are to be added.

Let C be the resulting measurement with error ΔC

C = a × b

ΔC/C = ±(Δa/a + Δb/b)

∴ ΔC = ±(Δa/a + Δb/b) × C

Example 1

Two quantities are measured as 25 ± 2 and 20 ± 1

Then the product of these two measurements is given by −

(25 ± 2) × (20 ± 1)

C = 25 × 20 = 500

ΔC/C = ±(2/25 + 1/20) = 40 + 25/500 = 13/100

∴ ΔC = 13/100 × C = 13/100 × 500 = 65

Hence,

product = 500 ± 65

Or

Product = 500 ± 13%

Example 2

Mass of a liquid is measured as 520 ± 5 and the volume is measured as 130 ± 2

Then density (d) is given by −

520 ± 5/130 ± 2

d_m = 520/130 = 4

Δd_m/d = 5/520 + 2/130 = 13/520

Δd_m = 13/100 × d = 13/520 × 4 = 0.1

Hence,

Density = 4 ± 0.1

Or

Density = 4 ± 2.5%