How to Calculate Uncertainty

Uncertainty Propagation Calculator

Use this calculator to determine the combined uncertainty when multiplying or dividing two measured quantities, each with its own absolute uncertainty. This method is based on the propagation of relative uncertainties for products and quotients.

Multiply (X * Y) Divide (X / Y)

Understanding and Calculating Measurement Uncertainty

In any scientific or engineering field, measurements are fundamental. However, no measurement is perfectly precise; every measurement carries some degree of doubt or "uncertainty." Understanding and quantifying this uncertainty is crucial for evaluating the reliability of experimental results, comparing different measurements, and making informed decisions.

What is Measurement Uncertainty?

Measurement uncertainty is a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand (the quantity being measured). It's not a mistake or an error in the sense of a blunder, but rather an inherent characteristic of the measurement process itself.

Sources of uncertainty can include:

• Instrument limitations: The precision of the measuring device.
• Environmental factors: Temperature, humidity, vibrations.
• Operator skill: How consistently a person takes readings.
• Definition of the measurand: How well the quantity being measured is defined.
• Sampling issues: Representativeness of the sample.

Expressing Uncertainty: Absolute vs. Relative

Uncertainty can be expressed in two primary ways:

1. Absolute Uncertainty (ΔX): This is expressed in the same units as the measured quantity. For example, if a length is measured as 10.0 cm with an absolute uncertainty of ±0.1 cm, it means the true value likely lies between 9.9 cm and 10.1 cm. It's often denoted as X ± ΔX.
2. Relative Uncertainty: This expresses the uncertainty as a fraction or percentage of the measured value. It's calculated as (ΔX / X). For the example above, the relative uncertainty would be (0.1 cm / 10.0 cm) = 0.01 or 1%. Relative uncertainty is useful for comparing the precision of measurements of different magnitudes.

Propagation of Uncertainty

Often, the quantity we are interested in (let's call it Z) is not measured directly but is calculated from other measured quantities (X, Y, etc.) that each have their own uncertainties. This process of determining the uncertainty of the calculated quantity from the uncertainties of the input quantities is called propagation of uncertainty.

The rules for propagating uncertainty depend on the mathematical relationship between the quantities. Our calculator focuses on two common scenarios: multiplication and division.

Rules for Multiplication and Division:

When a quantity Z is a product (Z = X * Y) or a quotient (Z = X / Y) of two independent measured quantities X and Y, the relative uncertainties combine in a specific way.

If:

• X has an absolute uncertainty ΔX
• Y has an absolute uncertainty ΔY

Then, the relative uncertainty of X is rX = ΔX / |X| and the relative uncertainty of Y is rY = ΔY / |Y|.

The combined relative uncertainty of Z (rZ) is given by:

rZ = sqrt( (rX)^2 + (rY)^2 )

Once you have rZ, the absolute uncertainty of Z (ΔZ) can be found by:

ΔZ = rZ * |Z|

Where Z is the calculated value (X * Y or X / Y).

Example Scenario: Calculating Area with Uncertainty

Imagine you are calculating the area of a rectangular plate. You measure its length (L) and width (W).

• Length (L) = 15.0 cm ± 0.2 cm
• Width (W) = 8.0 cm ± 0.1 cm

Here, X = L = 15.0, ΔX = ΔL = 0.2. And Y = W = 8.0, ΔY = ΔW = 0.1. The operation is multiplication (Area = L * W).

1. Calculate the value of Z (Area):
   Area = 15.0 cm * 8.0 cm = 120.0 cm²
2. Calculate relative uncertainties for L and W:
   rL = ΔL / L = 0.2 / 15.0 ≈ 0.013333
rW = ΔW / W = 0.1 / 8.0 ≈ 0.012500
3. Calculate the combined relative uncertainty for Area:
   rArea = sqrt( (0.013333)^2 + (0.012500)^2 )
rArea = sqrt( 0.00017777 + 0.00015625 )
rArea = sqrt( 0.00033402 ) ≈ 0.018276
4. Calculate the absolute uncertainty for Area:
   ΔArea = rArea * Area = 0.018276 * 120.0 cm² ≈ 2.1931 cm²

So, the area of the plate would be reported as 120.0 cm² ± 2.2 cm² (rounding ΔArea to one significant figure, which is common practice, and then matching the precision of the main value). The relative uncertainty would be 0.018276 * 100% = 1.83%.

Using the Uncertainty Propagation Calculator

Our calculator simplifies this process for you. Simply input your two measured values and their respective absolute uncertainties. Select whether you are multiplying or dividing these quantities, and the calculator will instantly provide:

• The Calculated Value (Z): The direct result of your chosen operation.
• The Absolute Uncertainty (ΔZ): The combined uncertainty in the same units as Z.
• The Relative Uncertainty (%): The absolute uncertainty expressed as a percentage of the calculated value.

This tool is invaluable for students, scientists, and engineers who need to quickly and accurately assess the uncertainty in their derived results.