Uncertainty Calculator using Partial Derivatives

This calculator demonstrates the principle of calculating uncertainty using partial derivatives, also known as propagation of uncertainty, for a function of two variables.

Results

z = 50.00 ± 2.24

Formula Used: δz = √((y·δx)² + (x·δy)²)

Intermediate Values:

Contribution from x: (∂z/∂x · δx)² = 1.00

Contribution from y: (∂z/∂y · δy)² = 4.00

What is Calculating Uncertainty Using Partial Derivatives?

Calculating uncertainty using partial derivatives, formally known as the propagation of uncertainty, is a fundamental method in experimental science and engineering to determine the uncertainty in a calculated quantity that depends on several other measured quantities. When you measure variables like length, time, or mass, each measurement has an associated uncertainty. If you use these measurements in a formula to calculate a new quantity (like area, speed, or density), the uncertainties of the initial measurements “propagate” to an uncertainty in the final result.

This method is crucial for anyone who needs to report the reliability of their results, including physicists, chemists, engineers, and data analysts. A common misunderstanding is to simply add the uncertainties of the input variables. However, the correct approach involves assessing how sensitive the final result is to changes in each input variable. This sensitivity is measured by the partial derivative of the function with respect to that variable. The individual contributions are then combined in quadrature (the square root of the sum of squares) to find the total uncertainty.

The General Formula for Uncertainty Propagation

For a function z = f(x, y), where x and y are independent measured variables with uncertainties δx and δy, the uncertainty in z, denoted as δz, is calculated using the following general formula:

δz = √[ ( (∂f/∂x)·δx )² + ( (∂f/∂y)·δy )² ]

This formula is the cornerstone of calculating uncertainty using partial derivatives. The term ∂f/∂x is the partial derivative of the function f with respect to the variable x, which quantifies how much z changes for a small change in x while y is held constant.

Variables in Uncertainty Propagation

Variable	Meaning	Unit (Auto-inferred)	Typical Range
z	The calculated final quantity.	Depends on function (e.g., m², m/s)	N/A
x, y	The independent measured variables.	Depends on measurement (e.g., m, s)	Positive real numbers
δx, δy	The absolute uncertainties in x and y.	Same as the corresponding variable	Small positive numbers
∂f/∂x, ∂f/∂y	Partial derivatives of the function f.	Unit of z / Unit of variable	Real numbers

Practical Examples

Example 1: Calculating the Area of a Rectangular Plate

Suppose you are measuring the area of a metal plate to determine if it meets specifications.

• Inputs:
  • Length (x): 10.0 meters
  • Uncertainty in Length (δx): 0.2 meters
  • Width (y): 5.0 meters
  • Uncertainty in Width (δy): 0.1 meters
• Function: Area (z) = x * y
• Partial Derivatives: ∂z/∂x = y = 5.0, ∂z/∂y = x = 10.0
• Calculation:
  • Area = 10.0 * 5.0 = 50.0 m²
  • δz = √[ (5.0 * 0.2)² + (10.0 * 0.1)² ] = √[ (1.0)² + (1.0)² ] = √[1 + 1] = √2 ≈ 1.41 m²
• Result: The area of the plate is 50.0 ± 1.41 m².

Example 2: Calculating Speed

An object’s speed is calculated from measurements of distance and time.

• Inputs:
  • Distance (x): 100 meters
  • Uncertainty in Distance (δx): 2 meters
  • Time (y): 20 seconds
  • Uncertainty in Time (δy): 0.5 seconds
• Function: Speed (z) = x / y
• Partial Derivatives: ∂z/∂x = 1/y = 1/20 = 0.05, ∂z/∂y = -x/y² = -100/400 = -0.25
• Calculation:
  • Speed = 100 / 20 = 5.0 m/s
  • δz = √[ (0.05 * 2)² + (-0.25 * 0.5)² ] = √[ (0.1)² + (-0.125)² ] = √[0.01 + 0.015625] = √0.025625 ≈ 0.16 m/s
• Result: The speed of the object is 5.0 ± 0.16 m/s.

How to Use This Uncertainty Calculator

This tool simplifies the process of calculating uncertainty using partial derivatives.

1. Select the Function: Choose the mathematical formula that relates your two measured variables (x and y) from the dropdown menu.
2. Enter Measured Values: Input the values you measured for variable ‘x’ and variable ‘y’.
3. Enter Uncertainties: Input the corresponding absolute uncertainties (δx and δy) for your measurements. These should be positive values representing the standard error or tolerance.
4. Interpret the Results: The calculator instantly updates. The primary result shows the calculated value of ‘z’ along with its total propagated uncertainty (z ± δz).
5. Analyze Contributions: The intermediate values and the bar chart show how much each variable’s uncertainty contributes to the final uncertainty. This is useful for identifying the largest source of uncertainty in your experiment. For further analysis you can check out our guide on propagation of uncertainty formula.

Key Factors That Affect Uncertainty Propagation

• Magnitude of Partial Derivatives: If the result is highly sensitive to one variable (large partial derivative), that variable’s uncertainty will have a magnified effect.
• Magnitude of Input Uncertainties: Naturally, larger uncertainties in the input measurements will lead to a larger uncertainty in the final result.
• The Mathematical Function: Multiplication and division propagate relative uncertainties, while addition and subtraction propagate absolute uncertainties. This affects how errors combine.
• Correlation Between Variables: This calculator assumes the input variables are independent. If they are correlated (e.g., measuring length and width with the same miscalibrated ruler), a covariance term is needed, which can increase or decrease the final uncertainty.
• Instrument Precision: The quality and resolution of your measurement tools are a primary source of the initial uncertainties.
• Random vs. Systematic Errors: This method is best suited for combining random uncertainties. Systematic errors (biases) must be corrected for separately.

Frequently Asked Questions (FAQ)

1. Why do you add the uncertainties in quadrature (sum of squares)?

Assuming the measurement errors are random and independent, some errors will be positive and some negative, leading to partial cancellation. Adding in quadrature is the statistically correct way to combine independent random uncertainties.

2. What if my function has more than two variables?

The formula extends naturally. For a function f(x, y, w, ...), you add a ( (∂f/∂variable)·δvariable )² term inside the square root for each additional variable.

3. What is the difference between uncertainty and error?

Error is the difference between a measured value and the true value. Uncertainty is a quantification of the doubt about the measurement result. We can estimate uncertainty, but we can never know the true error exactly.

4. Do the units for the uncertainty have to match the value?

Yes. The absolute uncertainty (e.g., δx) must have the same units as the measured value (x). For example, if you measure length in meters, its uncertainty must also be in meters.

5. Can I use relative uncertainties instead?

The general partial derivative formula uses absolute uncertainties. However, for functions involving only multiplication and division, it can be reformulated to use relative uncertainties (δx/x). For more info, see what is uncertainty propagation.

6. What does a partial derivative represent in this context?

It represents the sensitivity factor. It tells you how much the output (z) changes for every unit of change in one input (e.g., x), assuming all other inputs are constant.

7. When is this method not appropriate?

This method, which is a first-order Taylor series approximation, works best when uncertainties are small relative to the measured values. If uncertainties are very large, higher-order terms or methods like Monte Carlo simulation may be necessary.

8. What if my input uncertainties are correlated?

If errors in x and y are correlated, the general formula includes a covariance term: + 2(∂f/∂x)(∂f/∂y)σxy, where σxy is the covariance. This calculator assumes independence and omits this term.