Propagation of Error (Addition, Subtraction, Multiplication, and Division)

When performing calculations with measured values, it is important to understand how uncertainties in those measurements propagate through the calculations. This is known as the propagation of error. Each operation (addition, subtraction, multiplication, and division) has its own rule for how to combine the uncertainties of the individual measurements to find the uncertainty of the result.

Error Propagation in Addition and Subtraction

When adding or subtracting quantities, the absolute errors of the quantities are added to determine the absolute error of the result.

Formula

If $Q = A \pm B$ where $A$ and $B$ are quantities with uncertainties $\Delta A$ and $\Delta B$, respectively, the uncertainty in $Q$ is given by:

$$ \Delta Q = \Delta A + \Delta B $$

Example

If $A = 100 \pm 2$ and $B = 150 \pm 3$, then $Q = A + B = 250$ with an uncertainty of $\Delta Q = 2 + 3 = 5$. Thus, $Q = 250 \pm 5$.

Error Propagation in Multiplication and Division

When multiplying or dividing quantities, the relative (or percentage) errors of the quantities are added to determine the relative error of the result.

Formula

If $Q = A \times B$ or $Q = \frac{A}{B}$ where $A$ and $B$ are quantities with relative uncertainties $\frac{\Delta A}{A}$ and $\frac{\Delta B}{B}$, respectively, the relative uncertainty in $Q$ is given by:

$$ \frac{\Delta Q}{Q} = \sqrt{\left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2} $$

Example

If $A = 100 \pm 2$ and $B = 150 \pm 3$, then $Q = A \times B = 15000$ with a relative uncertainty of $\Delta Q/Q = \sqrt{(2/100)^2 + (3/150)^2}$. Calculating this gives $\Delta Q/Q \approx 0.022$, so $\Delta Q \approx 15000 \times 0.022 = 330$. Thus, $Q = 15000 \pm 330$.

Table of Differences and Important Points

Operation	Formula for Uncertainty	Example Calculation	Result
Addition	$\Delta Q = \Delta A + \Delta B$	$A = 100 \pm 2$, $B = 150 \pm 3$	$Q = 250 \pm 5$
Subtraction	$\Delta Q = \Delta A + \Delta B$	$A = 200 \pm 4$, $B = 150 \pm 3$	$Q = 50 \pm 7$
Multiplication	$\frac{\Delta Q}{Q} = \sqrt{\left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2}$	$A = 100 \pm 2$, $B = 150 \pm 3$	$Q = 15000 \pm 330$
Division	$\frac{\Delta Q}{Q} = \sqrt{\left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2}$	$A = 300 \pm 6$, $B = 150 \pm 3$	$Q = 2 \pm 0.06$

Important Points to Remember

• When adding or subtracting, use absolute errors.
• When multiplying or dividing, use relative errors.
• The square root in the multiplication and division formula accounts for the Pythagorean sum of errors, as errors are considered to be independent and random.
• Always report the final result with the error term, which indicates the uncertainty in the measurement.
• When raising a measurement to a power, multiply the relative error by the power. For example, if $Q = A^n$, then $\frac{\Delta Q}{Q} = n \cdot \frac{\Delta A}{A}$.
• It is important to keep track of significant figures; the result should not have more significant figures than the least precise measurement.

Understanding the propagation of error is crucial in experimental sciences and engineering, as it provides a quantitative measure of the confidence in the results obtained from measured data.