On this page
How to use the calculator
- Type the formula with a name for the result, like
g = 4 pi^2 L / T^2. The name is optional. A reciprocal form such as1/f = 1/do + 1/diis solved for f for you. - A row appears for every quantity in the formula. Type the measured value, its uncertainty and a unit label. Switch a row to % when the uncertainty is a percentage, like a meter's "±0.5% of reading".
- Check the Read as line under the formula once: it shows how your typing was understood. Then read the result, the table of contributions and the working.
| You type | Meaning |
|---|---|
m g / x or m*g/x | m times g, divided by x |
mg | one quantity named mg (a hint appears under the formula) |
a / b c | a/(bc): a space binds tighter than /, as in physics papers. Type a / b * c for (a/b)·c |
theta, θ, rho, m1, v_0 | single quantities, shown as θ, ρ, m₁, v₀ |
pi, e | the constants π and e (use E or q for an energy or a charge) |
sqrt( ), ln( ), log( ), exp( ), sin( ), atan( ) | functions; log is base 10 and ln is natural |
6.674e-11 | a number in scientific notation |
The rad and deg switch decides how sin, cos and tan read an angle. In degree mode an angle's uncertainty is in degrees too, and the tool puts the π/180 factor into the derivative.
Links that open a problem
Copy link builds a link like /error-propagation-calculator/?f=4pi^2L/T^2&L=1.000,0.002&T=2.007,0.005, which opens with the formula and values filled in. Teachers can send one link per lab.
| Parameter | Example | Meaning |
|---|---|---|
| f | f=4pi^2L/T^2 | the formula |
| a quantity's name | L=1.000,0.002,m | value, uncertainty, optional unit. End the uncertainty with % for a percentage: L=1.000,0.2% |
| unit | unit=m/s^2 | unit label of the result |
| deg | deg=1 | angles in degrees |
| corr | corr=V,I,-0.36 | a correlation; separate several with ; |
| mc | mc=1 | open the Monte Carlo check |
| round | round=pdg | rounding rule: one, two or pdg |
A quantity that shares a name with a parameter gets an underscore: _f=0.25,0.01.
The general formula
Every measurement x has a standard uncertainty u(x): the standard deviation of the values you could reasonably have got. When a result q = f(x₁, x₂, ...) is computed from measured inputs, each input's uncertainty moves the result by about its sensitivity, the partial derivative ∂f/∂xᵢ, times u(xᵢ). For independent inputs these contributions add in quadrature:
u(q)² = (∂f/∂x₁·u(x₁))² + (∂f/∂x₂·u(x₂))² + ...
This is the law of propagation of uncertainty in the GUM (the international Guide to the Expression of Uncertainty in Measurement) and in NIST Technical Note 1297. It is a first-order formula: it replaces f by its tangent plane at the measured point. That is accurate when the uncertainties are small compared with how fast the derivatives change.
Squares add, rather than the terms themselves, because independent errors partly cancel: a long L is as likely to come with a short T as a long one. Adding the absolute terms instead, Σ|∂f/∂xᵢ|·u(xᵢ), gives the worst case or maximum error, an upper bound some courses still ask for. The tool shows both, plus the relative uncertainty u(q)/|q| and the expanded uncertainty U = 2u, which covers about 95% of outcomes when the result is close to normal.
The tool finds each partial derivative symbolically, checks its value against a numerical derivative at your point, and prints it twice in the working: with symbols, then with your numbers in place. For how numerical derivatives work, see numerical differentiation explained.
Shortcut rules and when they apply
Lab manuals teach special cases of the general formula. They are what it gives for simple shapes:
| Result | Rule |
|---|---|
| q = a + b or a − b | absolute uncertainties add in quadrature: u(q) = √(u(a)² + u(b)²) |
| q = c·a, c exact | u(q) = |c|·u(a) |
| q = a·b or a/b | relative uncertainties add in quadrature |
| q = aⁿ | the relative uncertainty is multiplied by |n| |
| q = ln a | u(q) = u(a)/|a| |
They assume independent inputs, and each input must appear once. In q = x − 2y, u(q) = √(u(x)² + 4u(y)²): y is one reading multiplied by 2, not two independent readings. A formula that mixes sums and products, like z = w x + y^2, needs the rules in stages, while the general formula does it in one step. With w = 4.52 ± 0.02 cm, x = 2.0 ± 0.2 cm and y = 3.0 ± 0.6 cm, the tool gives z = 18.04 cm² with u = 3.71 cm², reported as (18 ± 4) cm².
Worked examples
1. g from a simple pendulum
A pendulum of length L = 1.000 ± 0.002 m has a period T = 2.007 ± 0.005 s, and g = 4π²L/T².
- g = 4π²·1.000/2.007² = 9.80088 m/s²
- ∂g/∂L = 4π²/T² = 9.80088 s⁻², so the L term is 9.80088 × 0.002 = 0.01960 m/s²
- ∂g/∂T = −8π²L/T³ = −9.76669 m/s³, so the T term is −9.76669 × 0.005 = −0.04883 m/s²
- u(g) = √(0.01960² + 0.04883²) = 0.05262 m/s², a relative uncertainty of 0.537%
Reported: g = (9.80 ± 0.05) m/s². The table shows T carries 86% of u², because T is squared and its relative uncertainty (0.249%) already exceeds L's (0.200%). Timing ten swings and dividing by ten would shrink that term most. The shortcut agrees: √(0.200%² + (2 × 0.249%)²) = 0.537%.
2. Density of a metal cylinder
Mass m = 52.36 ± 0.05 g, diameter d = 2.540 ± 0.005 cm and height h = 3.82 ± 0.01 cm give ρ = 4m/(πd²h) = 2.70507 g/cm³ with u = 0.01305 g/cm³, reported as (2.705 ± 0.013) g/cm³. The uncertainty keeps two figures because it starts with 1. The table is the useful part: d contributes 66.6% of u², h 29.5% and m only 3.9%. The balance is good enough. A micrometer for the diameter would help most, because d enters squared, so its 0.197% relative uncertainty counts as 0.394%.
3. Resistance from meters with percentage errors
A voltmeter reads 12.0 V ± 0.5% and an ammeter 0.250 A ± 1%. R = V/I = 48.0 Ω, with relative uncertainty √(0.5² + 1²)% = 1.118%, so u(R) = 0.537 Ω and R = (48.0 ± 0.5) Ω. Switch both rows to % and type the percentages as the meter's specification gives them.
Correlated inputs
Two inputs are correlated when their errors tend to move together, for example when both come from the same instrument reading. The formula then gains a covariance term for each pair, + 2·r·(∂f/∂xᵢ·uᵢ)(∂f/∂xⱼ·uⱼ), where the correlation coefficient r runs from −1 to 1. Press Add a correlation, pick the two quantities and type r.
Example H.2 of the GUM finds a resistance from R = (V/I)·cos φ with V = 4.9990 ± 0.0032 V, I = 19.6610 ± 0.0095 mA, φ = 1.04446 ± 0.00075 rad and the correlations r(V, I) = −0.36, r(V, φ) = 0.86, r(I, φ) = −0.65. The tool (with I entered in amperes) gives R = 127.732 Ω and u = 0.0700 Ω, matching the 0.071 Ω in the GUM and the 0.07 Ω from NIST's Uncertainty Machine. Leave the correlations out and u grows to 0.194 Ω: here the correlation terms cancel most of the variance, and the table lists them as negative shares.
Why run the Monte Carlo check
The general formula trusts the tangent. When the formula curves noticeably across ±u, the linear answer drifts. The Monte Carlo check runs the experiment numerically instead: it draws 100,000 sets of inputs from normal distributions with your values, uncertainties and correlations, pushes each set through the formula, and reports the mean, the standard deviation and the 95% interval between the 2.5% and 97.5% points, with a histogram. This is the method of the GUM's Supplement 1. When the two methods agree, the linear result is safe to report; when they do not, a note under the histogram says so.
A ball launched at v = 10.0 ± 0.1 m/s and θ = 45 ± 5° lands at R = v²·sin 2θ/g = 10.19 m, with g = 9.81 m/s² taken as exact. At 45° the range is at its maximum, so ∂R/∂θ = 0 and the linear method says the angle adds nothing: u = 0.20 m, all from v. The Monte Carlo check disagrees: mean 10.04 m, standard deviation 0.29 m, 95% interval 9.36 to 10.53 m. Any error in the angle can only shorten the throw, and a tangent cannot see that.
The NIST Uncertainty Machine manual has a laboratory case: the viscosity of a solution from a falling-ball timer, μ = μ_C(ρ_B − ρ_M)/(ρ_B − ρ_C)·t_M/t_C, with 10% and 15% uncertainties in the two fall times. The linear result is 5.69 mPa·s with u = 1.03 mPa·s. The Monte Carlo check gives a mean of 5.82, a standard deviation of 1.12 and a 95% interval of 4.05 to 8.40 mPa·s, skewed upward because t_C sits in the denominator. These are the values the manual reports.
Reporting the result with the right figures
An uncertainty is itself an estimate, good to one or two figures. The usual lab rule: round the uncertainty to one significant figure, or two if its first digit is 1, then round the value to the same decimal place. So z = 12.0349 cm with u = 0.153 cm is reported as (12.03 ± 0.15) cm, and g = 9.80088 m/s² with u = 0.05262 m/s² becomes (9.80 ± 0.05) m/s².
The rounding menu also offers always one figure, always two (the GUM recommends no more than two), and the Particle Data Group rule, which reads the first three digits of u: 100 to 354 keeps two figures, 355 to 949 keeps one, and 950 to 999 rounds up to 1000 with two. Under that rule 0.827 ± 0.367 becomes 0.8 ± 0.4. Round only at the end: the tool keeps full precision through every step and shows the unrounded numbers beside the rounded ones. Very large or small results use a shared power of ten, like (1.91 ± 0.13) × 10⁻⁴.
Common mistakes
- Angles in degrees. The derivative of sin θ is cos θ per radian. With θ = 53 ± 2°, the 2° must enter as 0.0349 rad; using 2 makes the angle term 57 times too big. Degree mode adds the π/180 factor for you. More in degrees vs radians on a calculator.
- Adding uncertainties by habit. The plain sum is the worst case, not the standard uncertainty. Quote the quadrature value unless your course asks for the maximum error.
- Mixing units. Units here are labels. If L is in cm and g should come out in m/s², convert L first.
- A range used as a standard uncertainty. A scale's half-division, or a specification of ±a, is a limit. The GUM divides such a half-range by √3 when every value inside it is equally likely, so ±0.5 mm becomes u = 0.29 mm.
- Correlated inputs treated as independent. That can overstate u, as in the GUM example above, or understate it.
Print working opens every section and prints the tool alone, ready to staple into a lab notebook.
Embed the Error Propagation Calculator on your website
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Frequently asked questions
What is the formula for propagation of uncertainty?
For a result q = f(x₁, x₂, ...) with independent inputs, u(q)² is the sum of (∂f/∂xᵢ · u(xᵢ))² over all inputs. Correlated pairs add a term 2·r·(∂f/∂xᵢ·uᵢ)(∂f/∂xⱼ·uⱼ). This first-order law is the one in the GUM and NIST Technical Note 1297.
Do I add uncertainties or add them in quadrature?
For independent random errors, add the squares and take the square root. Adding the terms directly gives the worst-case maximum error, which is always larger. For the pendulum example the quadrature value is 0.053 m/s² and the worst case is 0.068 m/s².
How many significant figures should an uncertainty have?
Usually one, or two when the first digit is 1, and the value is rounded to the same decimal place. A result of 12.0349 with an uncertainty of 0.153 is written as 12.03 ± 0.15. The GUM allows at most two figures.
How do I propagate the uncertainty of an angle?
The derivative of a trig function is per radian, so convert the angle's uncertainty to radians first: 2° is 0.0349 rad. In degree mode this calculator does the conversion inside the derivative, so you can type the angle and its uncertainty in degrees.
Why does the Monte Carlo result differ from the formula?
The formula uses the tangent at your measured point. When the function curves across the range of the inputs, as the range of a projectile does near 45°, the samples spread differently and the mean shifts. Then report the Monte Carlo standard deviation and 95% interval, or reduce the input uncertainties.
What is percent uncertainty?
It is the relative uncertainty u(x)/|x| times 100. A reading of 12.0 V with a 0.5% uncertainty has u = 0.06 V. Switch a row to % to type it that way; the result shows its relative uncertainty in percent too.