Error Propagation Calculator
Combine individual measurement uncertainties (σ) using standard error propagation rules for sums, products, and weighted means.
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About error propagation in surveying
Every measurement carries an uncertainty. When multiple measurements are combined (added, subtracted, multiplied, averaged), the resulting uncertainty follows specific rules derived from statistics. Surveyors use error propagation to predict the uncertainty of computed quantities: a traverse length depends on the sum of individual segments, a coordinate difference depends on two station positions, and so on.
For independent measurements with standard deviations σ₁, σ₂, ..., σₙ, the combined standard deviation for a simple sum or difference is σ = √(σ₁² + σ₂² + ... + σₙ²). For products and ratios, relative errors combine the same way. For weighted means (multiple observations of the same quantity), the combined precision improves: σ = 1/√(Σ 1/σᵢ²).
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Common use cases
Computing expected accuracy of a traverse from segment accuracies before fieldwork.
Combining GNSS baselines into a network and predicting station coordinate accuracy.
Averaging multiple observations of the same angle or distance to determine best-estimate precision.
Computing error budget for a surveying project before bidding to ensure specifications are achievable.
Frequently asked questions
Why do errors add in quadrature, not linearly?
Because individual errors are usually independent and randomly distributed around zero. Squaring and adding then square-rooting (the quadrature formula) gives the correct statistical combination. Linear addition would assume systematic errors.
What confidence levels mean in surveying?
1σ = 68% of measurements fall within this range. 2σ (≈1.96σ exact) = 95%. 3σ (≈2.58σ exact) = 99%. Survey specifications often require 2σ for acceptance.
When should I use weighted mean?
When you have multiple independent observations of the same quantity with different precisions. Weights are 1/σ². Poor observations contribute less to the final value.
Does this apply to GNSS?
Yes. GNSS baseline accuracy grows with baseline length (ppm), and multi-session combinations reduce uncertainty following these same rules.
Related tools
Use our RTK accuracy calculator for GNSS-specific error analysis and traverse closure for classifying finished surveys.