Error Propagation Calculator
Find the uncertainty of a formula result from measured values and their uncertainties. Shows each measurement’s contribution through partial derivatives, the root sum of squares and the maximum error, and a value ± uncertainty rounded to significant digits.
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Enter a formula with the measured values and their uncertainties to get the uncertainty of the result. The tool also shows how much each measurement contributes to it.
Supported: + − × ÷ ^, parentheses, π and e, sin cos tan asin acos atan exp ln log sqrt abs. log is base 10 and ln is natural. Variables are single letters (subscripts such as T2 or R_1 are allowed); Δt and θ also work. Angles are in radians.
Try an example
Enter the value and uncertainty (±) of each measured quantity in the formula.
Enter a formula to see the result and its uncertainty.
References
- NIST TN 1297 Appendix A: law of propagation of uncertainty (A-3)
The basis for combining independent input uncertainties with partial derivatives (root sum of squares).
- JCGM 100:2008 (GUM) 7.2.6: number of digits
The guideline that an uncertainty is usually adequate to two significant digits.
- Vern Lindberg, “Uncertainties and Error Propagation”
The source of the maximum-error method, the rule “two digits when the leading digit is 1”, and worked examples with answers.
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Error Propagation Calculator
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How to use
- 1
Enter the formula and the measured quantities
Type a formula such as g = 4π²L/T², then enter the value and uncertainty (±) of each measured quantity. Uncertainties can be absolute (same unit) or in percent.
- 2
Check the result and its uncertainty
Both the root sum of squares and the maximum error are shown. Choose the one your course or lab manual asks for.
- 3
Find the cause with contributions and working
A larger share of the variance means that quantity dominates the uncertainty. The working shows the substitution and partial derivatives.
Features
- Formulas with + − × ÷ ^, parentheses, π and e, trigonometric, exponential, logarithmic and square-root functions, with an optional name on the left
- Exact partial derivatives by automatic differentiation, with each quantity’s contribution and share of the variance
- Both the root sum of squares and the maximum error, reported with the method you choose
- Value ± uncertainty rounded to automatic, one or two digits, with an optional unit
- Working for substitution, partial derivatives, contributions and rounding, plus advice on reading the result
- Runs entirely in your browser; inputs are not sent or stored
Use cases
Lab report uncertainties
From measurements such as a pendulum or Ohm’s law, get the uncertainty of the result and decide which measurement to improve.
University physics and chemistry labs
Enter the formula and uncertainties to see contributions and working without computing partial derivatives by hand.
Estimating instrument accuracy
If an instrument’s accuracy is given in percent, convert it to an absolute uncertainty and estimate the result’s accuracy.
Details
If x and y carry uncertainties Δx and Δy, the uncertainty of the result f(x, y) can be estimated with the partial derivatives ∂f/∂x and ∂f/∂y. Combining independent uncertainties as a root sum of squares is the standard approach (NIST TN 1297, equation A-3). The upper bound, when every uncertainty pushes the result the same way, is the sum of |∂f/∂x|Δx (the maximum error).
Products, quotients and powers are easier to picture as relative uncertainties (Δf/|f|). For R = V/I, a 1% relative uncertainty in V and 2% in I give a relative uncertainty of √(1² + 2²) ≈ 2.2% in R. The measurement with the largest partial derivative affects the result most.
In a difference (ΔT = T2 − T1), the relative uncertainty grows as the two values get closer. The difference of 35.2 ± 0.1 and 34.9 ± 0.1 is 0.30 ± 0.14, a relative uncertainty of about 47%. In such cases, consider measuring the difference directly.
The tool parses the formula and computes exact partial derivatives by automatic differentiation. Enter angles in radians (convert a measured uncertainty in degrees with × π/180). The reported value rounds the uncertainty to one or two digits and rounds the value to the same decimal place. The default follows the common rule: two digits if the leading digit is 1, otherwise one digit.
Statistics, Sig Figs & Error Propagation
Calculate mean and unbiased variance, round to JIS significant figures, fit linear regressions, and propagate experimental uncertainties.
FAQ
What is error propagation?
It estimates how uncertain a result is when it is calculated from measured values that each carry an uncertainty. The uncertainty of f(x, y) is combined from the uncertainty of each measurement multiplied by its partial derivative.
Should I report the root sum of squares or the maximum error?
Follow the method your course or lab manual specifies. The root sum of squares is the standard way to combine independent uncertainties. The maximum error is a safe upper bound for the case where every uncertainty pushes the result the same way, so it is usually larger.
How many digits should the uncertainty have?
Many lab manuals use “two digits if the leading digit is 1, otherwise one digit”, which is the default here. The GUM (JCGM 100:2008, clause 7.2.6) states that quoting an uncertainty to about two significant digits is usually sufficient.
In which unit should I enter angle uncertainties?
Angles inside trigonometric functions are in radians, so enter the angle and its uncertainty in radians. Convert a degree uncertainty with × π/180.
Why does the uncertainty of a difference get large?
When two close values are subtracted, the absolute uncertainties stay about the same but the difference is small, so its relative uncertainty grows. For example, 35.2 ± 0.1 minus 34.9 ± 0.1 gives 0.30, a relative uncertainty of about 47%.
Are my inputs saved?
The calculation runs only in your browser. Inputs are neither sent nor stored, and they disappear when you leave the page.
Related guide
GuideGuide to Error Propagation: Formulas, Partial Derivatives, and Uncertainty Reporting
Master error propagation and uncertainty synthesis for science and engineering lab reports. Covers addition/subtraction, multiplication/division, power rules, general partial derivatives, root-sum-of-squares vs worst-case bounds, and GUM significant figure rounding with worked examples.
GuideHow to Use Ohm’s Law: Voltage, Current, Resistance and Power
Learn to use Ohm’s law V = IR to find voltage, current and resistance, plus the three power formulas P = VI, P = I²R and P = V²/R — with a 6-pair lookup table, comparison charts, and worked LED resistor examples.
GuideHow to Count Significant Figures and Apply the Rounding Rules
Count significant figures correctly, handle zeros, choose between half-up and half-even rounding, and apply the multiplication/division and addition/subtraction rules with worked examples.
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Verified: Textbook worked examples (w·x → 9.0 ± 0.9; x/y → 0.67 ± 0.15 and 0.7 ± 0.2; x·cos θ → 1.20 ± 0.12 and ± 0.13; A·sin y → 1.120 ± 0.008), pendulum sensitivities, agreement with finite differences, and parsing and error cases in unit tests. Example values were checked by independent hand calculation and by a Monte Carlo run for the density example
Did you know?
Independent uncertainties are usually combined as a root sum of squares. Uncertainties of 0.3 and 0.4 combine to 0.5, not the simple sum of 0.7.