Understanding Error Propagation

Calculating Uncertainty in Calculated Results.

What is Error Propagation?

Error Propagation (or propagation of uncertainty) is the study of how uncertainties in measured quantities are transferred to a final result that is calculated from those quantities.

Whenever we measure a physical quantity (like length, mass, or time), there is always some degree of uncertainty or error. When we use these uncertain measurements in a calculation, the uncertainty 'propagates' to the final answer.

The goal of error propagation is to determine the uncertainty in the calculated result based on the uncertainties of the initial measurements.

Example: If you measure the length and width of a rectangle, each with a small uncertainty, what is the resulting uncertainty in the calculated area? Error propagation provides the answer.

Rule 1: Addition and Subtraction

When adding or subtracting measured quantities, the uncertainty in the result is found by adding the absolute uncertainties in quadrature (using squares and square roots).

If a result q = x + y or q = x - y, and the uncertainties are δx and δy, the uncertainty in q (δq) is:

δq = √[(δx)² + (δy)²]

Note: You always add the uncertainties in quadrature, even when subtracting the quantities themselves.

Example:You measure two lengths: L₁ = (10.0 ± 0.1) cm and L₂ = (5.0 ± 0.2) cm. The total length is L = 15.0 cm. The uncertainty is δL = √[(0.1)² + (0.2)²] ≈ 0.22 cm. So, the final result is (15.0 ± 0.22) cm.

Rule 2: Multiplication and Division

When multiplying or dividing measured quantities, it's the fractional (or relative) uncertainties that are added in quadrature.

If a result q = xy or q = x/y, the fractional uncertainty in q is given by:

(δq / |q|) = √[(δx / x)² + (δy / y)²]

To find the final absolute uncertainty (δq), you first calculate the fractional uncertainty and then multiply it by the calculated value of q.

Example:To find the area of a rectangle with L = (10.0 ± 0.1) cm and W = (5.0 ± 0.2) cm: Area = 50 cm². Fractional uncertainty = √[(0.1/10)² + (0.2/5)²] ≈ 0.0412. Absolute uncertainty δA = 0.0412 * 50 ≈ 2.06 cm². So, Area = (50.0 ± 2.1) cm².

Real-World Application: Experimental Science

Error propagation is a fundamental part of all experimental sciences and engineering.

Physics and Chemistry Labs: When calculating a value like density from measured mass and volume, scientists must propagate the errors from the balance and the graduated cylinder to report a meaningful uncertainty in the final density.

Engineering: When designing a component, engineers must consider the manufacturing tolerances (uncertainties) of its parts. Error propagation helps them calculate the potential uncertainty in the final assembly's dimensions or performance.

Data Analysis: It allows scientists to state how confident they are in their results. A result of '10 ± 2' is far less certain than a result of '10.00 ± 0.01'.

Example:A pharmacist compounding a medication must account for the uncertainties in their measurements to ensure the final dosage is within a safe and effective range.

Key Summary

**Error Propagation** determines the uncertainty in a result calculated from uncertain measurements.
For addition/subtraction, add **absolute uncertainties** in quadrature.
For multiplication/division, add **fractional uncertainties** in quadrature.
Reporting uncertainty is essential for expressing the confidence and validity of an experimental result.

Practice Problems

Problem: You measure the initial and final temperatures of a water bath as T_initial = (20.5 ± 0.2)°C and T_final = (35.0 ± 0.2)°C. What is the change in temperature (ΔT) and its uncertainty?

The change is ΔT = T_final - T_initial. Use the rule for addition/subtraction to find the uncertainty.

Solution: ΔT = 35.0 - 20.5 = 14.5°C. The uncertainty is δ(ΔT) = √[(0.2)² + (0.2)²] = √[0.08] ≈ 0.28°C. So, ΔT = (14.5 ± 0.28)°C.

Problem: You measure the voltage across a resistor as V = (9.0 ± 0.1) V and the current through it as I = (2.0 ± 0.05) A. Calculate the resistance R = V/I and its uncertainty.

First calculate R. Then use the rule for multiplication/division to find the fractional uncertainty, and finally the absolute uncertainty.

Solution: R = 9.0 V / 2.0 A = 4.5 Ω. Fractional uncertainty (δR/R) = √[(0.1/9.0)² + (0.05/2.0)²] ≈ √[(0.0111)² + (0.025)²] ≈ 0.0274. Absolute uncertainty δR = 0.0274 * 4.5 Ω ≈ 0.12 Ω. So, R = (4.5 ± 0.12) Ω.

Frequently Asked Questions

What is the difference between systematic and random errors?

Random errors are unpredictable fluctuations in measurements (e.g., noise in an electronic sensor), and they can be reduced by averaging multiple measurements. Systematic errors are consistent, repeatable errors (e.g., a miscalibrated scale), and they cannot be reduced by averaging. Error propagation formulas primarily deal with random errors.

Why do we add errors in quadrature (using squares and square roots)?

This method is used for independent random errors. It's a statistical approach that recognizes it's unlikely that the maximum possible errors in each measurement will both occur at the same time and add up perfectly. The quadrature sum gives a more probable and realistic estimate of the combined uncertainty.

What is the rule for raising a measurement to a power?

If q = xⁿ, the rule is a simplification of the multiplication rule. The fractional uncertainty is simply multiplied by the absolute value of the exponent: (δq / q) = |n| * (δx / x).

Error Propagation Calculator

Error Propagation Calculator