Calculating the Uncertainty

Uncertainty Propagation Calculator

Calculate the uncertainty of a derived quantity from the uncertainties of its measured components.

Understanding Uncertainty in Measurements

In scientific and engineering fields, no measurement is perfectly precise. Every measurement carries some degree of uncertainty, which reflects the range within which the true value of the measured quantity is expected to lie. Understanding and quantifying this uncertainty is crucial for evaluating the reliability and validity of experimental results.

What is Uncertainty?

Uncertainty is a quantification of the doubt about the measurement result. It tells us how much we trust our measurement. It's often expressed as a range, for example, 10.0 ± 0.1 cm, meaning the true value is likely between 9.9 cm and 10.1 cm.

Absolute vs. Relative Uncertainty

• Absolute Uncertainty (ΔX): This is the actual amount of uncertainty in the measurement, expressed in the same units as the measured quantity. For example, in 10.0 ± 0.1 cm, 0.1 cm is the absolute uncertainty.
• Relative Uncertainty (ΔX/X): This expresses the uncertainty as a fraction or percentage of the measured value. It's often more useful for comparing the precision of different measurements. For 10.0 ± 0.1 cm, the relative uncertainty is 0.1/10.0 = 0.01 or 1%.

Why Calculate Uncertainty?

When we perform experiments, we often measure several quantities and then use these measurements to calculate a derived quantity. For instance, we might measure the mass and volume of an object to calculate its density. If our initial measurements have uncertainties, then the calculated density will also have an uncertainty. Propagating these uncertainties correctly ensures that the final result accurately reflects the precision of the entire experiment.

• Scientific Rigor: It's a fundamental aspect of good scientific practice to report not just a value, but also its uncertainty.
• Comparison of Results: It allows for meaningful comparison between different experiments or theoretical predictions. If two results' uncertainty ranges overlap, they are considered consistent.
• Identifying Limiting Factors: By seeing which input uncertainty contributes most to the final uncertainty, researchers can identify areas where measurement techniques need improvement.

Rules for Propagating Uncertainty (Multiplication and Division)

The method for propagating uncertainty depends on the mathematical operation used to combine the measured quantities. This calculator focuses on multiplication and division, which follow a specific rule for combining relative uncertainties.

If a quantity Z is calculated from two measured quantities A and B, where Z = A × B or Z = A ÷ B, and A has an absolute uncertainty ΔA and B has an absolute uncertainty ΔB, then the rule for propagating uncertainty is:

(ΔZ / Z) = (ΔA / A) + (ΔB / B)

This means that the relative uncertainty of the product or quotient is the sum of the relative uncertainties of the individual quantities. Once you have the relative uncertainty of Z (ΔZ / Z), you can find the absolute uncertainty of Z (ΔZ) by multiplying it by the calculated value of Z:

ΔZ = Z × ((ΔA / A) + (ΔB / B))

Example: Calculating Density and its Uncertainty

Let's say we measure the mass and volume of a substance to determine its density. We have the following measurements:

• Mass (A): 100.0 g ± 0.5 g (So, A = 100.0, ΔA = 0.5)
• Volume (B): 10.0 cm³ ± 0.1 cm³ (So, B = 10.0, ΔB = 0.1)

Density (Z) is calculated as Mass / Volume (Z = A ÷ B).

Step-by-Step Calculation:

1. Calculate the nominal value of Z (Density):
   Z = A / B = 100.0 g / 10.0 cm³ = 10.0 g/cm³
2. Calculate the relative uncertainty of A (Mass):
   ΔA / A = 0.5 g / 100.0 g = 0.005
3. Calculate the relative uncertainty of B (Volume):
   ΔB / B = 0.1 cm³ / 10.0 cm³ = 0.01
4. Calculate the relative uncertainty of Z (Density):
   (ΔZ / Z) = (ΔA / A) + (ΔB / B) = 0.005 + 0.01 = 0.015
5. Calculate the absolute uncertainty of Z (Density):
   ΔZ = Z × (ΔZ / Z) = 10.0 g/cm³ × 0.015 = 0.15 g/cm³

So, the density would be reported as 10.0 ± 0.15 g/cm³, with a relative uncertainty of 1.5%.

Use the calculator above to quickly perform these calculations for your own experimental data!