Measurement uncertainty calculator: free uncertainty budget with Welch–Satterthwaite

A measurement uncertainty calculation turns everything that affects a result into one number: list the contributions, convert each into a standard uncertainty, combine them as a root sum of squares, and multiply by a coverage factor k. This calculator does that with the GUM method, including the step most spreadsheets skip: effective degrees of freedom (Welch–Satterthwaite) and a k from the t distribution, so a budget built on three readings doesn’t get k = 2 when it needs 2.28.

The calculations run in your browser and nothing is sent anywhere. They are checked by 105 automated tests, most of them against worked examples from EA-4/02, the GUM, UKAS M3003 and JCGM GUM-5 (validation). New to the topic? Start with our guide to measurement uncertainty.

How to use the calculator

1. Start from an example or a blank budget. The calculator opens with an official worked example, EA-4/02 S3 (a 10 kΩ resistor). Pick another one and press Load example, or press Start a blank budget.
2. Enter the result. The measured value y (optional), its unit and the coverage probability. The default, 95.45 %, gives exactly k = 2 when the degrees of freedom are large.
3. Add one row per contribution.
   • Type A: paste the repeated readings, one per line. The calculator gives the mean, the standard deviation s, the standard uncertainty of the mean s/√n and ν = n − 1. Numbers use the US format: a dot for decimals and, optionally, commas for thousands (1,234.5); a value like 0,25 is flagged rather than guessed.
   • Type B: choose what you have (a certificate U with its k, limits ±a, a resolution step…) and enter the value. Leave ν blank for the usual assumption of infinite degrees of freedom.
   • Sensitivity coefficient c: how much the result changes per unit of that input. Use 1 when the input is already in the unit of the result.
4. Read the budget. Each row shows its standard uncertainty and its contribution |c|·u. The bars show each contribution’s share of u_c², so you can see which one to work on first.
5. Read the result. u_c, ν_eff, k and U, plus a result statement you can copy. Download CSV saves the whole budget for your records or for Excel.

How the calculator works

The calculator follows the GUM (JCGM 100:2008) for independent inputs and a linear model, the method used in calibration laboratories and set out for them in EA-4/02. The measurement uncertainty guide explains each idea in depth; this is the short version.

Type A contributions come from statistics on repeated readings. For n readings, the standard uncertainty of their mean is s/√n, with n − 1 degrees of freedom (GUM 4.2.3, G.3.3).

Type B contributions come from any other information: certificates, specifications, drift history, handbooks. Each is converted into a standard uncertainty with a divisor that depends on what you know about it:

The calculator then multiplies each standard uncertainty by its sensitivity coefficient c_i and combines the results as the square root of the sum of squares, the law of propagation of uncertainty for uncorrelated inputs (GUM 5.1.2; EA-4/02 §4).

Effective degrees of freedom and the coverage factor

A combined standard uncertainty built partly on a few readings is itself uncertain. The GUM handles this in four steps (GUM G.6.4; NIST TN 1297, B.3; EA-4/02, Annex E):

1. Give each contribution its degrees of freedom: n − 1 for Type A. For a Type B input, ν is usually taken as infinite when its limits were chosen so that values outside them are very unlikely (GUM G.4.3). If a certificate states finite degrees of freedom, use them (UKAS M3003, B.5).
2. Combine them with the Welch–Satterthwaite formula: ν_eff = u_c⁴ / Σ (c_iu_i)⁴ / ν_i (GUM G.4.1, eq. G.2b; NIST TN 1297, eq. B-1). Contributions with infinite ν add nothing.
3. If ν_eff is not a whole number, truncate it to the next lower integer. The GUM allows interpolating or truncating, and EA-4/02 truncates (GUM G.6.4; EA-4/02, E2). The calculator truncates by default; under Options you can use the exact value instead, as JCGM GUM-5 does (§2.4.1.4).
4. Take k = t_p(ν_eff), the Student t value for the chosen coverage probability, and U = k·u_c.

The calculator computes t directly rather than looking it up, and agrees with 167 of the 168 printed values of GUM Table G.2; the other differs by 0.01.

Example: EA-4/02 S12. A water meter’s relative error is measured in three runs: 0.0003, 0.0005 and 0.0022. The standard uncertainty of their mean is 0.60 × 10⁻³ with ν = 2, and the error of a single run adds 0.68 × 10⁻³ (normal, ν = ∞). Combined, u_c = 0.91 × 10⁻³, ν_eff = 10.3, truncated to 10, and k = 2.28 at 95.45 %, not 2 (EA-4/02, S12.13–S12.15). Load it in the calculator to see each step.

When does k = 2 do? The GUM notes that it is often adequate when there are enough well-behaved contributions of similar size and ν_eff is greater than about 10, and that for ν_eff around 11 it gives an interval about 10 % too narrow at 95 % (GUM G.6.6). NIST uses k = 2 by convention (NIST TN 1297, §6.5). Computing ν_eff costs nothing and tells you which case you are in.

When one rectangular contribution dominates

The t-based k assumes the result is close to normally distributed. That breaks down when a single rectangular contribution, typically the resolution of a low-resolution display, is much larger than the rest (GUM G.6.5; EA-4/02, S9.11). UKAS gives a rule of thumb: the rectangular term dominates when the combined uncertainty of all other terms is less than about 0.70 times its own, at 95.45 % (UKAS M3003, C.4).

When that happens the calculator shows a warning and offers, as an option, a second k computed for a rectangular distribution combined with a normal remainder (UKAS M3003, C.6–C.7; GUM G.6.5). The t-based k stays in the result until you tick the box. It agrees with UKAS M3003 table C.7 to within 0.01 (26 of 26 values checked). In the limit of a purely rectangular result, k for 95 % is 0.95 × √3 = 1.65, which is what EA-4/02 uses for a handheld multimeter at 100 V (EA-4/02, S9.12–S9.14). Load the S9 example to see the warning.

Worked example: a 10 kΩ standard resistor (EA-4/02 S3)

This is the example the calculator opens with. A 10 kΩ resistor is calibrated by substitution against a reference resistor of the same nominal value, with a 7½-digit multimeter. The model is R_X = (R_S + δR_D + δR_TS) · r_C · r − δR_TX, so the two ratios r_C and r enter with a sensitivity coefficient of about 10 000 Ω (EA-4/02, S3.2–S3.9).

What the budget shows:

• The drift of the reference standard is the largest contribution (5.8 mΩ, about half of u_c²), larger than the reference’s own calibration (2.5 mΩ). A shorter calibration interval or a better drift history would reduce U more than a better certificate.
• Five readings barely matter here. The repeatability of the ratio contributes 0.7 mΩ. Its ν = 4 does not lower ν_eff much (about 77 000), so k stays at 2.00.
• u_c = 8.33 mΩ and U = 2 × 8.33 ≈ 17 mΩ, reported as (10 000.178 ± 0.017) Ω (EA-4/02, S3.9–S3.11). The calculator gives the same values.

Reporting the result

Accreditation policy for calibration certificates asks for (Global ACI-TECH-1-009, §5.2–5.3):

• the result as y ± U, with units;
• the coverage factor and the coverage probability, with a short note saying that U is the standard uncertainty multiplied by k for a coverage probability of approximately 95 %;
• U to at most two significant digits, with rounding done after all the calculations.

The calculator rounds U to two significant digits and the measured value to the same decimal place (GUM 7.2.6), and writes the note for you. Some EA-4/02 examples print U with one digit, such as 0.002 for S12; with two digits the calculator shows 0.0021.

To check whether that U is small enough for a tolerance, take it to the TUR calculator. If your results sit near the limits, see guard banding and decision rules.

Validation

The math lives in a separate library and is checked by 105 automated tests: most compare it with published worked examples and the printed t table, and the rest are consistency checks. The main examples:

For S9 and S11, EA-4/02 derives k from the shape of the dominant distributions (1.65 and 1.81); the calculator’s k for a dominant rectangular term gives 1.70 and 1.82, and the same rounded U.

Assumptions and limits

• Independent inputs. The calculator does not handle correlations. Correlated inputs need covariance terms (GUM 5.2).
• Linear model. Sensitivity coefficients are entered by you, as the first-order partial derivatives of your model. Strongly non-linear models, or results with an asymmetric distribution, may need the Monte Carlo method of JCGM 101. NIST’s free Uncertainty Machine and Sandia’s SunCal support it.
• Welch–Satterthwaite is an approximation. It approximates the distribution of the result by a t distribution (GUM G.4.1–G.4.2).
• Type B with ν = ∞ is a common assumption, valid when the limits are conservative (GUM G.4.3). If you doubt a Type B value, you can give it finite degrees of freedom: if you think it is reliable to about 25 %, ν ≈ 8 (GUM G.4.2).
• One dominant term at a time. The warning looks at the largest rectangular contribution, with the UKAS rule of thumb set for 95.45 %; at 99 % treat it as a guide. Two comparable dominant rectangular terms form a trapezoidal distribution, which EA-4/02 treats separately (S10.13).
• Your budget, your responsibility. The calculator does the arithmetic; deciding which contributions matter, and documenting why, is part of your procedure.