Use this **propagation of uncertainty calculator** to determine the final measurement uncertainty when combining two independent measurements (Length and Width) through multiplication (Area). This calculator uses the relative uncertainty method for multiplicative relationships.
Propagation of Uncertainty Calculator
Calculated Area (A): —
Uncertainty in Area ($\delta A$): —
Propagation of Uncertainty Formula: Area
For multiplication and division ($A = L \times W$ or $A = L / W$), the relative uncertainties add in quadrature:
Step 1: Calculate Area (A)
A = L \times W
Step 2: Calculate Relative Uncertainty ($\delta A / A$)
\frac{\delta A}{A} = \sqrt{ \left(\frac{\delta L}{L}\right)^2 + \left(\frac{\delta W}{W}\right)^2 }
Step 3: Calculate Absolute Uncertainty ($\delta A$)
\delta A = A \times \frac{\delta A}{A}
Variables
- Length (L): The measured value of the first dimension. Must be positive.
- Uncertainty in Length ($\delta L$): The absolute error or uncertainty associated with the Length measurement. Must be non-negative.
- Width (W): The measured value of the second dimension. Must be positive.
- Uncertainty in Width ($\delta W$): The absolute error or uncertainty associated with the Width measurement. Must be non-negative.
Related Calculators
What is Propagation of Uncertainty?
Propagation of Uncertainty is a mathematical process used to estimate the uncertainty in a final calculated quantity that is derived from multiple measured variables, each having its own inherent uncertainty. When you measure two quantities, $L$ and $W$, each with a small error ($\delta L$ and $\delta W$), any quantity calculated from them, like Area ($A=L \times W$), will also carry an error ($\delta A$).
The standard method assumes that the input errors are independent and random. The formula uses partial derivatives to determine how much the uncertainty in each input variable contributes to the total uncertainty of the output. This is a crucial step in all experimental sciences and engineering to ensure the reliability of results.
How to Calculate Uncertainty Propagation (Example)
- Input Values: Length $L = 10.0 \pm 0.1$ units and Width $W = 5.0 \pm 0.2$ units. Thus, $\delta L = 0.1$ and $\delta W = 0.2$.
- Calculate Base Result: $A = 10.0 \times 5.0 = 50.0$ square units.
- Calculate Relative Uncertainty in L and W: $$\frac{\delta L}{L} = \frac{0.1}{10.0} = 0.01$$ $$\frac{\delta W}{W} = \frac{0.2}{5.0} = 0.04$$
- Sum Relative Uncertainties in Quadrature: $$\frac{\delta A}{A} = \sqrt{ (0.01)^2 + (0.04)^2 } = \sqrt{0.0001 + 0.0016} = \sqrt{0.0017} \approx 0.04123$$
- Calculate Absolute Uncertainty: $$\delta A = A \times 0.04123 = 50.0 \times 0.04123 \approx 2.06$$
- Final Result: The Area is $50.0 \pm 2.1$ square units.
Frequently Asked Questions (FAQ)
Why do we use the square root of the sum of squares?
This method, called adding in quadrature, is used because we assume the errors ($\delta L$ and $\delta W$) are independent and random. Squaring removes negative signs, and the square root ensures the final unit is correct. It prevents the errors from simply adding up linearly, which would be an overestimation for random errors.
Can this formula be used for addition or subtraction?
No. For addition ($A=L+W$) or subtraction ($A=L-W$), the absolute uncertainties are added in quadrature: $\delta A = \sqrt{ (\delta L)^2 + (\delta W)^2 }$. The relative uncertainty method is specific to multiplication and division.
What happens if the base measurement (L or W) is zero?
If the base measurement is zero, the relative uncertainty formula is undefined (division by zero). In such a non-physical scenario, the calculation cannot proceed, and a different method (or better measurement) is required.
What is the difference between relative and absolute uncertainty?
Absolute uncertainty ($\delta A$) has the same units as the measured quantity ($A$). Relative uncertainty ($\delta A / A$) is unitless and often expressed as a percentage, representing the size of the error relative to the quantity’s magnitude.