# Error Propagation Calculator - Uncertainty, Monte Carlo, Weighted Fit

> Propagate uncertainty through any formula: enter values ± errors to get the result, partial derivatives, error budget, Monte Carlo check and weighted line fit.

- URL: https://qilingtools.com/en/physics/error-propagation
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Type a formula such as g = 4π²L/T² and give each measured quantity its uncertainty. The tool differentiates the formula symbolically, combines the contributions in quadrature (independent errors) or as a worst-case linear sum, and rounds the result properly, e.g. 9.80 ± 0.05. A Monte Carlo run checks whether the linear approximation holds, and a second tab fits a straight line to data with error bars, giving slope and intercept uncertainties and χ²/ν.

## How to use

1. Type a formula or load an example: pendulum g, density, resistance, kinetic energy, projectile range
2. Enter each variable's value and uncertainty (absolute or %), ticking “angle” for variables in degrees
3. Choose quadrature or worst-case combination and a rounding rule, then read the rounded result
4. Check the partial derivatives and each variable's share, and compare with the Monte Carlo histogram
5. For a line fit, switch to “Weighted line fit” and paste x, y, σ_y columns

## Features

- Symbolic partial derivatives, cross-checked against numeric central differences
- Quadrature and worst-case combination with an error-budget bar chart
- Uncertainty rounded to 1–2 significant figures (PDG rule and others) plus concise notation
- Monte Carlo propagation with histogram, percentile intervals and skewness
- Weighted least-squares line fit: slope, intercept, covariance, χ²/ν, p-value, residual plot
- One-click examples: pendulum, density, resistance, kinetic energy, projectile, lens

## FAQ

### When should I add uncertainties in quadrature and when linearly?

Use quadrature, u_f² = Σ(∂f/∂xᵢ·uᵢ)², when the input errors are independent, for example a length from a ruler and a time from a stopwatch; this is the GUM standard and what most lab manuals expect. The linear sum u_f = Σ|∂f/∂xᵢ|uᵢ assumes every error pushes the result the same way at once, so it is an upper bound, useful for conservative error limits or when errors may be strongly correlated and you cannot estimate the correlation.

### How many significant figures should the uncertainty have?

Usually one or two. The Particle Data Group rule looks at the first three digits of the uncertainty: 100–354 keep two, 355–949 keep one, and 950–999 round up to 1000 and keep two. The value is then rounded to the same decimal place, so 9.8009 ± 0.0526 becomes 9.80 ± 0.05. The tool also offers “two digits if the first is 1, otherwise one” and “always two”.

### Monte Carlo disagrees with linear propagation — which one is right?

Linear propagation uses only first derivatives, treating the formula as a straight line near the measured values. When the relative uncertainties are large or the formula curves strongly (1/x, eˣ, square roots near zero), the output distribution becomes skewed and its centre and width drift from the linear estimate. Monte Carlo is then the better description, so report its 68% or 95% percentile interval; the tool flags nonlinearity from the width ratio, mean shift and skewness.

### How do I read χ²/ν in the weighted line fit?

χ² = Σ(yᵢ − a − bxᵢ)²/σᵢ² and ν is the number of points minus the number of fitted parameters. With realistic error bars and a correct straight-line model, χ²/ν is close to 1 and the p-value is not tiny. Much larger than 1 means the error bars are too small or the data curve away from a line (look at the residuals); much smaller than 1 suggests overestimated error bars. If the model is right and the errors were simply underestimated, tick the option that scales parameter errors by √(χ²/ν).

### Can it compute the mean and standard deviation of repeated measurements?

This tool focuses on propagating known input uncertainties through a formula and on straight-line fits. For the mean, standard deviation and confidence interval of repeated readings, and for outlier tests such as Q and Grubbs, use the site's Lab data & error analysis tool; for significant-figure rules and arithmetic use the Significant Figures Calculator.

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