The error propagation calculator above answers the question every measured calculation raises: if each quantity going in has an uncertainty, how uncertain is the number coming out? You type the formula you genuinely use rather than picking from a list of preset shapes, enter up to three measured quantities with their uncertainties, and the tool returns the combined standard uncertainty, the relative uncertainty, an expanded interval at whatever coverage factor you choose, and a breakdown of how much each input contributed.
Arb Digital built this page around the contribution breakdown because that is the part people never compute by hand and the part that changes what you do next. A combined uncertainty is a number to quote. The contribution split is a decision: it tells you which instrument to replace, which measurement to repeat, and which one to stop worrying about. Uncertainties combine as a sum of squares, so a component half the size of the largest contributes only a quarter as much variance, and improving it is usually wasted effort.
What This Error Propagation Calculator Does
It applies the general law of propagation of uncertainty to an arbitrary function of up to three variables. Rather than restricting you to sums, products and powers, it evaluates your formula and its partial derivatives numerically, which means the same engine handles a square root, a logarithm, a trigonometric term or any combination of them. The closed forms for the common cases fall out of it automatically, so you do not have to decide in advance which rule applies.
It assumes the input uncertainties are independent and uncorrelated. That assumption is stated here, in the tool, and in the results, because it is where most real propagation goes wrong. If two of your quantities were measured with the same instrument, calibrated against the same standard, or read off the same drifting thermometer, their errors move together and the simple quadrature formula is no longer correct.
This is a different job from our percent error calculator, which compares a single measured value against a known true value to express how far off you were. That is a statement about accuracy after the fact. This page is a statement about uncertainty before you have any true value to compare against, propagated through a calculation you are about to report.
How to Use It
- Write the formula. Use x, y and z as your variable names. Multiplication needs an explicit asterisk, and powers use the caret, so a cylinder volume is pi*x^2*y with x as the radius and y as the height.
- Enter each value with its uncertainty. The uncertainty should be a standard uncertainty, meaning roughly a one-standard-deviation figure, not a worst-case bound.
- Leave unused variables inert. If your formula only uses x and y, set z to 1 with an uncertainty of 0 and it will contribute nothing.
- Choose a coverage factor. Leave k at 1 to report the standard uncertainty. Set it to 2 for an interval that covers about 95 percent of a normal distribution.
- Read the contribution bars. They tell you where the uncertainty in your result is actually coming from.
The Formula and How It Is Calculated
For a function q = f(x, y, z) with independent inputs, the combined standard uncertainty is u(q) = √[(∂f/∂x)2u(x)2 + (∂f/∂y)2u(y)2 + (∂f/∂z)2u(z)2]. Each partial derivative is a sensitivity coefficient: it says how much the result moves per unit change in that input. Multiply it by that input's uncertainty and you have the contribution, and the contributions add as squares. NIST's guidance on combining uncertainty components gives this expression, along with the simplified forms for a weighted sum and for a product of powers.
The calculator obtains each derivative by central difference, evaluating the formula at x + h and x − h with a step scaled to the magnitude of x, and dividing the difference by 2h. This is accurate to well beyond the precision anyone quotes an uncertainty to, and it works for functions where writing the derivative by hand would be tedious or error-prone.
The familiar shortcuts are special cases of the same expression. For a sum or difference, the derivatives are 1 and −1, so the absolute uncertainties add in quadrature. For a product or quotient, differentiating and dividing through by q makes the relative uncertainties add in quadrature. For a power q = xn, the relative uncertainty is multiplied by |n|, so squaring a quantity doubles its percentage uncertainty and taking a square root halves it. Note the consequence for a difference: subtracting two nearly equal numbers keeps the absolute uncertainty roughly the same while shrinking the result, so the relative uncertainty can explode.
Independence Is the Assumption That Breaks
Every formula on this page assumes the errors in x, y and z are uncorrelated. In a laboratory that assumption is often false in ways that are easy to miss. Two lengths measured with the same steel rule share the rule's calibration error. Two masses read from the same balance share its zero offset. Two concentrations prepared from the same stock solution share whatever error was in the stock. In each case the errors move together, and the exact expression needs an extra covariance term for each correlated pair.
The direction of the mistake depends on the algebra. If the errors are positively correlated and the quantities are added, ignoring the correlation understates the combined uncertainty, sometimes badly. If those same positively correlated quantities are subtracted, the shared error partly cancels and ignoring the correlation overstates the uncertainty. Neither outcome is safe, and the NIST/SEMATECH e-Handbook's discussion of propagation of error considerations makes the case that a top-down estimate from repeated whole-process measurements handles covariances properly and catches sources you never thought to list.
There is a related trap inside the formula itself. If the same physical measurement appears twice, write it as one variable. The tool differentiates with respect to x wherever x appears, which correctly treats those occurrences as the same quantity. Entering the same measurement as both x and y instead tells the tool they are two independent readings, and for an expression like x − y that turns an exact zero into a spurious uncertainty.
Linearisation and When First Order Is Not Enough
The propagation law is a first-order Taylor expansion. It replaces the function with its tangent plane at your measured point and propagates the uncertainty through that straight-line approximation. This is excellent when the uncertainties are small relative to the curvature of the function, which covers the overwhelming majority of practical work.
It degrades in two identifiable situations. The first is a strongly curved function evaluated over a wide uncertainty, such as an exponential with a large uncertainty in its argument, where the tangent line badly misrepresents the true spread and the result is no longer symmetric. The second is a near-singular point: dividing by a quantity whose uncertainty is a substantial fraction of itself, or taking a logarithm of a value that could plausibly be near zero. Watch for a relative uncertainty above roughly 20 percent on any quantity in a denominator or inside a nonlinear function, and treat the answer as indicative rather than exact. A Monte Carlo simulation is the standard fallback when the linear approximation is inadequate.
Rounding, Significant Figures and How to Report the Result
An uncertainty is normally quoted to one or two significant figures, and the value is then rounded to the same decimal place as the uncertainty. Reporting 6.0234871 ± 0.25 is internally inconsistent: the uncertainty says the second decimal is already unreliable, so the digits after it are noise. Write 6.02 ± 0.25, or 6.0 ± 0.3 if you prefer one figure on the uncertainty.
Round only at the end. Rounding intermediate quantities and then propagating introduces an error of its own that is invisible in the final figure. Our significant figures calculator handles the digit-counting rules, and our scientific notation converter is useful when the result and its uncertainty span very different orders of magnitude. Always state whether the number you quote is a standard uncertainty or an expanded one, and if expanded, give the coverage factor — an unlabelled interval is ambiguous by a factor of two.
Where the Input Uncertainties Come From
Propagation is only as good as the uncertainties you feed it, and those come from two distinct routes. A Type A evaluation estimates uncertainty statistically from repeated observations: measure the same quantity several times, take the standard deviation of the readings, and divide by the square root of the number of repeats to get the standard uncertainty of the mean. Our standard deviation calculator and standard error calculator perform exactly those two steps.
A Type B evaluation uses everything else: a manufacturer's specification, a calibration certificate, or the resolution of the instrument. Where a specification gives a bound rather than a standard deviation — a digital display with a half-count resolution, say — the usual convention is to treat the value as uniformly distributed across that bound and divide the half-width by √3. The two types are combined identically once converted to standard uncertainties, which is the whole reason for converting them first.
A Worked Example You Can Check Yourself
The defaults are the rectangle-area case: x = 3.0 ± 0.1 as the length, y = 2.0 ± 0.05 as the width, and the formula x*y. The area is 6.00. The partial derivative with respect to x is y = 2, and with respect to y is x = 3. The two contributions are therefore 2 × 0.1 = 0.2 and 3 × 0.05 = 0.15.
Adding those in quadrature gives u = √(0.22 + 0.152) = √(0.04 + 0.0225) = √0.0625 = 0.25 exactly. So the area is 6.00 ± 0.25, a relative uncertainty of 4.17 percent. Cross-check with the product rule: the relative uncertainties are 0.1/3 = 3.33 percent and 0.05/2 = 2.50 percent, and √(3.332 + 2.502) = 4.17 percent, which matches.
Now read the contributions. The variance from x is 0.04 out of 0.0625, which is 64 percent; y supplies the other 36 percent. If you could halve the uncertainty in y to 0.025, the combined uncertainty would fall only to 0.2136 — a 15 percent improvement. Halving the uncertainty in x instead takes it to 0.1803, a 28 percent improvement. Same effort, nearly double the payoff, and the bars told you which one to pick before you did any of the arithmetic.
Arb Digital's analysts carry uncertainty through a calculation instead of quoting the last figure a spreadsheet happened to print.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Adding uncertainties directly instead of in quadrature — straight addition assumes every error happens at its worst in the same direction at once, which overstates the combined figure substantially.
- Mixing absolute and relative rules — absolute uncertainties add in quadrature for sums, relative uncertainties add in quadrature for products, and swapping the two gives an answer that is wrong by whatever the magnitudes happen to be.
- Assuming independence without checking — measurements sharing an instrument, a calibration or a stock solution have correlated errors that the quadrature formula does not cover.
- Entering a worst-case bound as a standard uncertainty — convert a bound to a standard uncertainty first, typically by dividing a uniform half-width by the square root of three.
- Quoting more digits than the uncertainty supports — round the value to the same decimal place as the uncertainty, and round only once, at the end.
Related Free Tools From Arb Digital
Compare a measurement against a known value with the percent error calculator, estimate a Type A uncertainty with the standard deviation calculator, scale it to the mean with the standard error calculator, check the squared spread with the variance calculator, get the digit rules right with the significant figures calculator, or tidy an awkward magnitude with the scientific notation converter. The free online tools hub lists every measurement tool we publish.
Frequently Asked Questions
Because independent errors are as likely to partly cancel as to reinforce each other. Adding them directly assumes every error takes its largest value in the same direction simultaneously, which is a worst case rather than a typical one.
It means the error in one quantity carries no information about the error in another. Measurements sharing an instrument, a calibration standard or a common preparation step usually violate this, and then an extra covariance term is required.
Multiply the relative uncertainty by the absolute value of the exponent. Squaring a quantity doubles its percentage uncertainty, cubing it triples it, and taking a square root halves it.
A multiplier applied to the standard uncertainty to give an interval with a stated level of confidence. A factor of two gives roughly 95 percent coverage when the combined result is approximately normally distributed.
The absolute uncertainty stays about the same while the result itself becomes small, so the uncertainty as a fraction of the answer grows sharply. Designing the measurement to avoid such a subtraction is usually better than improving the instruments.
When the function is strongly curved across the range of your uncertainties, or when a denominator could plausibly approach zero. A relative uncertainty above roughly twenty percent inside a nonlinear term is a warning sign, and a Monte Carlo simulation is the usual alternative.
No. Carry full precision through the whole calculation and round once at the end, matching the value to the decimal place of its uncertainty. Rounding partway adds an error the final figure cannot show.
This page explains a measurement calculation for educational purposes only. It is not a substitute for a formal uncertainty budget prepared under an accredited method, and the standard uncertainties it combines are only as good as the values you supply.
