A quantity (x) is measured with N (sample size) distinct values and associated uncertainty σₓ (its standard error). When calculating monetary values obtained by multiplying, for example, the unit price p by the physical quantity x, the combined uncertainty (margin of error) of the product of the two variables (m = p·x) must be considered:

(1) m = (p ± σₚ) × (x ± σₓ)

A first-order Taylor expansion (error-propagation formula) gives the variance:

(2) (σₘ)² = (x·σₚ)² + (p·σₓ)² + 2·p·x·σₚₓ

where σₚₓ = Cov(p, x) is the covariance between p and x.

Dividing equation (2) by m² = (p·x)² yields the squared coefficient of variation (CV):

(3) (σₘ / m)² = (σₚ / p)² + (σₓ / x)² + 2·σₚₓ / (p·x)

The coefficient of variation is useful for comparing measurement reliability: the estimate with the smallest CV is the most reliable.

If the errors of p and x are uncorrelated (σₚₓ ≈ 0), equation (3) simplifies to:

(4) (σₘ / m)² = (σₚ / p)² + (σₓ / x)²

or equivalently:

(5) σₘ / m = √[ (σₚ / p)² + (σₓ / x)² ]

For details, see Ifan Hughes and Thomas Hase (2010, pp. 44, 95) or Taylor (2022, pp. 23, 34, 53).

Apropos Transfer Pricing

Transfer pricing often involves ad-hoc adjustments to the selected uncontrolled profit indicator—e.g., “adjusting” a reported ROS (return on sales)—without evaluating the resulting standard errors. Yet every adjustment adds algebraic operations that propagate uncertainty. Thus, the adjusted ROS can become less reliable than the original (unadjusted) profit indicator.

Using a less reliable (higher variance) adjusted profit indicator violates the scientific principle of minimizing measurement error. Avoiding error propagation is essential to producing credible, reproducible (defensible) estimates of the selected statistical measure such as profit indicator.

Mathematical Appendix

Here is a derivation of the error propagation formulae for multiplying two variables (p and x)—offering more insight than the typical textbook approach, which only states the final equations.

For a statistical measure or value involving two variables: m = p x, the error propagation formula is:

(1a) σₘ² = (∂m/∂p)² σₚ² + (∂m/∂x)² σₓ² + 2(∂m/∂p)(∂m/∂x) σₚₓ.

Since ∂m/∂p = x and ∂m/∂x = p, this simplifies to:

(2a) σₘ² = (x σₚ)² + (p σₓ)² + 2 (p x) σₚₓ.

Dividing σₘ² by m² = (p x)² yields:

(3a) (σₘ/m)² = (σₚ/p)² + (σₓ/x)² + 2 σₚₓ/(p x).

If p and x are uncorrelated (σₚₓ ≈ 0), this reduces to:

(4a) (σₘ/m) = √[(σₚ/p)² + (σₓ/x)²].

Price and quantity may be related through Alfred Marshall’s trope about supply and demand (two blades of a pair of scissors). Thus, Cov(p, x) = σₚₓ ≠ 0 and the applicable error propagation formula is (3a) instead of equation (4a).

References

Hughes, Ifan, and Thomas Hase. Measurements and Their Uncertainties: A Practical Guide to Modern Error Analysis. Oxford University Press, 2010.

Taylor, John. Introduction to Error Analysis: Uncertainties in Physical Measurements (3rd edition). University Science Books, 2022.