Law of Propagation of Uncertainty

Law of Propagation of Uncertainty
The Law of Propagation of Uncertainty is the cornerstone of modern metrology, based on the Guide to the Expression of Uncertainty in Measurement (GUM). It is used to determine how doubts (uncertainties) in input measurements affect the final result of a calculated quantity.

Let's start by understanding what "uncertainty" is.

In real life, nothing is measured with absolute precision. If you measure a table with a ruler, you might say it measures 1 meter, but in reality, it measures 1 meter ± 1 millimeter. That "± 1 millimeter" is uncertainty: it's the reasonable margin of doubt we have about our measurement. Although the GUM (Guide to the Expression of Uncertainty in Measurement) defines it more technically as: "A non-negative parameter that characterizes the dispersion of the values attributed to a measurand, based on the information used," colloquially we can say that it's the doubt we have about the measurement taken, or a measure of the quality of the measurement. The smaller the uncertainty, the higher the quality of the measurement (better method, better conditions, and better measuring equipment). If you want to learn more about uncertainty, visit our other article with a practical example here.

What is the measurand?

According to the VIM (International Vocabulary of Metrology), a measurand is: "The quantity to be measured." It seems like a simple definition, but the key word is "to be measured." The measurand is the ideal concept, the specific quantity we want to know under perfect conditions. In a calibration, we usually want to measure the indication error or the correction factor, but in a normal measurement without comparison, what we usually want to measure is a quantity like length, area, density, etc. And here a problem arises:

We almost never measure something directly and that's it; we usually use that data to calculate something else.

Example: You want to know the area of a plot of land, so you measure the length and the width.
Your measuring tape has an uncertainty in the length and an uncertainty in the width because you took two measurements.
What is the uncertainty of the total area? You can't simply add the uncertainties of the sides, because area is a product of two measurements. This is where the Law of Propagation comes in.

To understand how uncertainty propagates, we need to know three key concepts:

1. The Mathematical Model
This is the formula that describes the behavior of your measurement. For example: Speed = Distance / Time. If the formula changes, the way uncertainty propagates also changes.

2. Sensitivity Coefficients
This is a tool that measures the "cause and effect" relationship between the parts of a system and its final result. In short, it tells us how much the result (measurand) is affected by each influencing variable in the mathematical model. The sensitivity coefficient indicates how much the output changes if one of the inputs varies slightly, while keeping the others constant.

Mathematically, it is the partial derivative of the function with respect to that variable:

3. Combination
In the world of uncertainty, doubts don't simply add up algebraically (1+1=2). They are added using the Pythagorean Theorem (the square root of the sum of the squares). Why? Because doubts can be upward (+) or downward (-), and squaring them ensures that they always add to the total uncertainty, never subtract from it. Each of these uncertainties is known as an associated uncertainty, and there must be one for each variable in your mathematical model.

Steps for Uncertainty Assessment

For an uncertainty budget to be technically valid, it must follow this sequence:

Analysis of Sources: Identify what affects each input variable (systematic effects, random effects, resolution, drift, etc.).
Quantification (Type A and Type B):
- Type A: Based on statistics (standard deviation of repeatability).
- Type B: Based on external information (calibration certificates, manuals, technical judgment, etc.).
Calculation of Standard Uncertainties (u): Bring all sources to a common standard deviation or according to their distribution. For example, for a uniform distribution, divide the resolution by 2√3 or √12.
Combination: Apply the law of propagation of uncertainties (the first formula).
Expanded (U): Multiply by the coverage factor k (usually k=2 for a 95.45% confidence level).

If you need more information on this, click here.

Practical Example:

Imagine you measure the length (L) and width (W) of a plate to obtain the area (S = L * W).

Model: S = L * W

Sensitivity Coefficients:

Applying the law of propagation of uncertainty, it becomes as follows:

Where uc(S) is the combined uncertainty of the area.

A and L are the sensitivity coefficients, which in this case are Length and Width (resulting from the partial derivatives).

uL and uA are the uncertainties associated with each measurement (resolution, repeatability, calibration, etc.).

Worked example with numbers

You measure a plate: length L = 250.00 mm with one caliper and width A = 120.00 mm with another caliper (each with its own calibration certificate). The area is S = L × A = 30,000 mm². The sensitivity coefficients come from differentiating the model: cL = ∂S/∂L = A = 120 mm and cA = ∂S/∂A = L = 250 mm.

Uncertainty sources (the same ones you would enter in the calculator below):

• Calibration of each caliper: U = 0.02 mm with k = 2 → u = 0.010 mm (custom divisor = 2).

• Resolution of 0.01 mm: uniform distribution (2√3) → u = 0.0029 mm.

• Repeatability (standard deviation of the mean): length u = 0.005 mm, width u = 0.004 mm (normal, divisor 1).

Each contribution is (c · u)². Length: 1.44 + 0.12 + 0.36. Width: 6.25 + 0.52 + 1.00. Sum = 9.69 mm⁴.

uc(S) = √9.69 = 3.11 mm² and U = 2 × 3.11 = 6.2 mm² (k = 2, ≈ 95 %). Result: S = (30,000 ± 6) mm².

The width weighs more even though it was measured just as well: its sensitivity coefficient (250) is larger than the length’s (120). To improve the result, tackle the largest contribution first, which here is the calibration of the width caliper.

Frequently asked questions

What is a sensitivity coefficient?

It is how much the result changes when one input changes by one unit; mathematically, the partial derivative of the model with respect to that input. If the model is a sum or difference, all coefficients are 1.

Which divisor should I use for each source?

Calibration certificate: U/k (usually k = 2). Resolution of a digital instrument: a/(2√3). Limits ± a with no other information (tolerance, manufacturer specification): a/√3 (rectangular). If values near the center are more likely: a/√6 (triangular). Repeatability (s/√n) is already a standard uncertainty: divisor 1.

Why multiply by k = 2?

The combined uncertainty uc is equivalent to one standard deviation (≈ 68 %). With k = 2 the expanded uncertainty U covers about 95 % of the reasonable values, which is what calibration certificates usually report.

What if two inputs are correlated?

The calculator’s formula assumes independent inputs. If two inputs share a source (for example, the same reference standard or the same instrument), you must add the GUM covariance term, or treat that common source as a single input.

When is the law of propagation not a good choice?

When the model is strongly non-linear around the measured value, or a single contribution dominates with a non-normal distribution. In those cases the GUM recommends the Monte Carlo method (GUM Supplement 1).

If you need to apply the calculation, we have provided a calculator for the law of propagation of uncertainty, considering the following: