Uncertainty Propagation Calculator

The value is g = 4π² × 0.9500 / 1.955² = 9.8127 m/s². T contributes 95.9% of u(g)² and L only 4.1%, so timing is what limits this measurement. Rounded to one significant figure, as many school courses ask, the result is g = (9.81 ± 0.05) m/s². With k = 2 the expanded uncertainty is ± 0.10 m/s².

In the What if? tab, halving u(T) brings u(g) down to 0.027 m/s², while halving u(L) only takes it to 0.050 m/s². Timing 10 swings and dividing by 10 spreads the stopwatch error over ten periods.

The add-percentages method gives 0.105% + 2 × 0.256% = 0.617% of 9.8127, so ± 0.061 m/s². It is larger because it adds the contributions in full instead of combining them in quadrature. The Monte Carlo check with 200,000 draws gives a standard deviation of 0.0513 m/s², so the formula method is fine here.

Open “Compare with an accepted value” and type 9.81. The difference is +0.0027 m/s², which is 0.053 standard uncertainties, so the result agrees with the accepted value.

The law of propagation of uncertainty

Propagation of uncertainty, also called error propagation, works out how the uncertainties of your measurements carry through a formula to its result.

Your result is y = f(x₁, …, xₙ), and each input xᵢ has a standard uncertainty u(xᵢ). The calculator multiplies each one by its sensitivity coefficient cᵢ = ∂f/∂xᵢ and combines the products in quadrature, as the GUM describes in section 5.1.2:

u(y)² = Σ (∂f/∂xᵢ · u(xᵢ))²

With correlated inputs it adds the cross terms of GUM 5.2.2, where r(xᵢ, xⱼ) is the correlation coefficient of two inputs:

u(y)² = Σ cᵢ² u(xᵢ)² + 2 Σᵢ<ⱼ cᵢ cⱼ r(xᵢ, xⱼ) u(xᵢ) u(xⱼ)

The partial derivatives are exact at your values. The formula is differentiated by automatic differentiation, not by nudging the inputs, and the Steps tab prints each derivative as a formula too.

Shortcuts you may know

• Sums and differences: add the absolute uncertainties in quadrature, u(a + b)² = u(a)² + u(b)².
• Products and quotients: add the relative uncertainties in quadrature, (u(y)/y)² = (u(a)/a)² + (u(b)/b)².
• Powers: y = xⁿ gives u(y)/|y| = |n| · u(x)/|x|.

The general law gives the same answers in these cases, and it also handles sin, ln, sqrt and any mix of them. For a pure product, quotient or power, or a pure sum or difference, the Steps tab also shows these rules at work.

Where the input uncertainties come from

• ± absolute: the number you enter is used as the standard uncertainty u.
• % of value: u = |x| × p/100. You can also type a % sign after any uncertainty.
• Expanded ±U ÷ k: a calibration certificate quotes an expanded uncertainty U with a coverage factor k, usually 2. Then u = U/k (GUM 4.3.3).
• Rectangular ±a (Type B, GUM 4.3.7): any value within ± a is equally likely, so u = a/√3. Use it for a stated tolerance.
• Triangular ±a (GUM 4.3.9): values near the middle are more likely, so u = a/√6.
• Resolution step: a display that moves in steps of δ hides up to half a step either way, so u = δ/√12 (GUM F.2.2.1). For a display showing 9.81 the step is 0.01.
• Readings: s/√n (Type A, GUM 4.2.3): the value is the mean of your n readings, and u = s/√n, where s is their standard deviation. It has n − 1 degrees of freedom. If every reading is the same, s is 0, but the value is not exact. The resolution of the instrument still limits it (GUM F.2.2.1). The page says so and asks for a Resolution step.
• Readings: half range: the value is the mean and u = (max − min)/2. This is a school rule for a handful of readings, not a GUM method. With 2 readings it equals s/√n, so it counts 1 degree of freedom, as s/√n does. From 3 readings on it is taken as it stands, without degrees of freedom. For a result from this input alone, k = 2 covers about 92% with 3 readings and more than 95% from 4 readings on.

Constants and exact numbers

π and e are built in and exact. Here e is Euler’s number, 2.718…, so use another name, such as q_e, for the electron charge. For a constant you look up, type it into the formula as a number, or keep it as a variable with uncertainty 0. If the value you use is rounded, as in g = 9.81 m/s², choose “Resolution step”. Enter the size of the last digit, here 0.01.

Expanded uncertainty and degrees of freedom

The expanded uncertainty is U = k·u (GUM 6.2.1). With k = 2 the interval y ± U covers about 95% of the likely values. That holds if the result is normally distributed and rests on many readings. With k = 3 it covers about 99.7%; the GUM calls that approximately 99% (6.3.3).

When an input is the mean of only a few readings, k = 2 covers less than 95%. The page then works out the effective degrees of freedom with the Welch–Satterthwaite formula (GUM G.4.1). Each input worked out as s/√n from n readings counts n − 1, and so does half the range of 2 readings. It shows the coverage your k really gives and the k that gives 95%, both from Student’s t-distribution. From 200 effective degrees of freedom on, the result is treated as normal, because the difference is under 1%. Inputs joined by a correlation count as one component with the fewest degrees of freedom among them.

Rounding

The uncertainty is rounded to two significant figures by default, the most GUM 7.2.6 recommends, and the value to the same decimal place. You can switch to one significant figure, as many school courses ask. A common lab rule keeps two figures only when the first one is 1. “Unrounded” shows 8 significant figures. The budget, the steps, “Copy steps” and “Copy as LaTeX” show each input with the digits you typed, so 0.9500 keeps its zeros. “Copy table” and “Download CSV” give plain numbers for a spreadsheet.

Add percentages (school method)

Many school courses add percentage uncertainties. For y = a·b/c, the percentage uncertainty of y is the sum of those of a, b and c. A power n counts n times. The calculator does the general version, Σ|cᵢ|·u(xᵢ), which matches that rule for products, quotients and powers. It is never smaller than the quadrature result, and it ignores correlations.

When the simple method fails: the Monte Carlo check

The law of propagation uses straight-line slopes at your values. If the formula bends noticeably within a few uncertainties, as y = x² does near x = 0, it can be far off.

The Monte Carlo check uses the method of GUM Supplement 1 (JCGM 101:2008). It draws a value for every input, puts each set through your formula and looks at the spread of the results. This is what it draws:

• ± absolute, % of value and Expanded ±U ÷ k: a normal distribution with standard deviation u (Supplement 1, 6.4.7).
• Rectangular ±a and Resolution step: a rectangular distribution (6.4.2). Triangular ±a: a triangular one (6.4.5).
• Readings: s/√n: a t-distribution with n − 1 degrees of freedom, centred on the mean and scaled by s/√n (6.4.9). With few readings it is wider than a normal one. With 2 or 3 readings it has no standard deviation, so the page then gives only the 95% interval.
• Readings: half range: a normal distribution with standard deviation (max − min)/2. With 2 readings it is drawn like Readings: s/√n, which it then equals. Supplement 1 has no rule for this school method.
• Correlated inputs are drawn together. For normal inputs this follows Supplement 1 (6.4.8). Correlated readings with the same number of readings share one joint t-distribution with n − 1 degrees of freedom; otherwise each keeps its own. For ±a inputs, each keeps its own shape and r is applied to the normal scores behind the draws. Supplement 1 gives no rule for that case.

The check reports the mean, the standard deviation and the 95% interval from the 2.5th to the 97.5th percentile (7.7). Draws where the formula has no value, such as the square root of a negative number, are left out, and the page says how many.

It is simpler than Supplement 1 in four ways:

• The number of draws is fixed at 200,000 or 1,000,000. Supplement 1 also describes a run that keeps drawing until the numbers are stable (7.9). With 200,000 draws the standard deviation is good to about 1%, or a few percent when an input rests on five or fewer readings. The ends of the interval are good to about 3% of the spread.
• It gives the interval with 2.5% of the draws on each side, not the shortest 95% interval.
• It judges the formula method with a plain rule instead of the validation test in clause 8. The same draws also go through the straight-line version of your formula, which is what the formula method uses. The two agree when three things hold. Their spreads differ by less than 10%. The centre has moved by less than a fifth of the spread. Neither end of the 95% interval has moved by more than 10% of its half-width. With inputs from readings, the spreads compared are the 95% half-widths and the centres are the medians.
• For several quantities read together n times, Supplement 2 (JCGM 102:2011) uses fewer degrees of freedom than n − 1. This page keeps n − 1, as Supplement 1 does for a single quantity.

When the spread or the centre disagrees, the page shows the Monte Carlo result as the one to quote. When only an end of the interval has moved, it keeps the formula result and adds the Monte Carlo interval. It says whether that interval is lopsided, wider or narrower.

When several inputs each rest on a few readings, the Monte Carlo interval is wider than value ± k·u, even for a straight-line formula. Supplement 1 treats the scatter of a few readings more cautiously than the GUM’s degrees-of-freedom formula. The page shows both intervals and says by how much they differ.

Sources

• JCGM 100:2008, Evaluation of measurement data: Guide to the expression of uncertainty in measurement (GUM), BIPM.
• JCGM 101:2008, Supplement 1 to the GUM: Propagation of distributions using a Monte Carlo method, BIPM.
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results (1994).

The correlated example uses the five sets of readings in GUM Annex H.2, Table H.2. From them the calculator gives R = (127.732 ± 0.071) Ω, the value and uncertainty printed in Table H.3. With only five sets of readings, 95% coverage needs k = 2.78, which gives ± 0.20 Ω.