Uncertainty Calculator
Master the propagation of error. Calculate combined uncertainties for complex lab formulas.
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Why Errors Propagate
Every measurement has a limit. When you combine imperfect measurements, their imperfections combine too—but not always simply.
Imagine measuring the length and width of a rectangle to find its area. If your length is "off" by a little bit, and your width is "off" by a little bit, how "off" is the area? That is the core question of Error Propagation.
In scientific contexts, we generally assume errors are random and independent. This means it's unlikely that both your length and width are wrong in the maximum possible way at the same time. One might be slightly high, while the other is slightly low, cancelling out some of the error. This is why we use the Quadrature Method (Adding squares) rather than simple addition.
The Rules of Propagation
Add absolute uncertainties in quadrature.δZ = √(δA² + δB²)
Add relative uncertainties in quadrature.δZ/Z = √((δA/A)² + (δB/B)²)
Multiply relative uncertainty by power |n|.δZ/Z = |n| × (δA/A)
Reporting Your Data
1. Match Precision: The result should never be more precise than its uncertainty.
12.345 ± 0.112.3 ± 0.1
2. One Sig Fig for Error: Uncertainties are estimates. Reporting ± 0.02341 is misleading.
Use ± 0.02
Frequently Asked Questions
How do you calculate uncertainty for addition and subtraction?
For Z = A ± B, you add the absolute uncertainties in quadrature: δZ = √(δA² + δB²). This assumes the errors are independent and random.
How do you calculate uncertainty for multiplication and division?
For Z = A × B or Z = A / B, you add the fractional (relative) uncertainties in quadrature: δZ/Z = √((δA/A)² + (δB/B)²). Then multiply by Z to get the absolute uncertainty.
What is the uncertainty power rule?
For Z = Aⁿ, the relative uncertainty is multiplied by the power |n|: δZ/Z = |n| × (δA/A). For example, if you square a value, you double its relative error.
Why do we use quadrature instead of just adding errors?
Adding errors directly (δZ = δA + δB) is the "worst-case scenario" method. Quadrature (Pythagorean sum) is statistically more probable for random, independent errors because it accounts for the likelihood that errors might cancel each other out.
What is relative uncertainty?
Relative uncertainty expresses the error as a fraction of the value (δA/A) or a percentage. It tells you the "quality" of the measurement regardless of the magnitude.
How many significant figures should uncertainty have?
Standard practice is to round uncertainty to 1 significant figure (e.g., ±0.03 g). If the leading digit is 1, some conventions allow 2 significant figures (e.g., ±0.14 mL).
Does a constant have uncertainty?
Exact numbers (like the "2" in 2πr) have zero uncertainty. Measured constants (like g = 9.81 m/s²) have uncertainty depending on their precision, but in many student labs, they are treated as exact relative to the rough experimental data.
What is the difference between accuracy and uncertainty?
Uncertainty quantifies the bound of confidence (precision) of the measurement. Accuracy quantifies how close the measurement is to the true value (error).
Can uncertainty be negative?
No. Uncertainty represents a magnitude of doubt, so it is always an absolute (positive) value.
What units does uncertainty have?
Absolute uncertainty has the same units as the original measurement. Relative uncertainty is unitless (dimensionless).
How does uncertainty propagate in a log function?
For Z = ln(A), δZ = δA / A. For Z = log₁₀(A), δZ = 0.434 × (δA / A).
What if errors are dependent?
If errors are correlated (dependent), you cannot use quadrature. You must add them linearly: δZ = δA + δB. This gives a larger, more conservative uncertainty.
How do I reduce uncertainty?
Use more precise instruments (smaller graduations), take more readings to reduce random error (standard error of the mean), and control environmental variables.
What is standard deviation vs uncertainty?
Standard deviation describes the spread of data points. The uncertainty of the mean is the Standard Error (SD / √N). Often, uncertainty is approximated as 1 SD or 2 SD (95% confidence).
Why is the power rule different?
Because A × A is perfectly correlated with itself. You can't treat it like A × B where errors might cancel. The errors always stack in the same direction, hence we multiply by n instead of using square roots.