Example: from five chamber readings to a reported result
Field Pro · updated 28 October 2026
Five readings from a vented ionisation chamber, taken on a warm day at low pressure, have to become one reported value with an expanded uncertainty. This example follows them through the air-density correction, the Type A evaluation and the combination with a certificate component, and shows two places where a hand calculation commonly goes wrong.
Inputs
Keep the default N-80 field and fill panels 03 and 04 in the workbench. The readings are in the instrument's own unit; only ratios matter here.
| Panel | Field | Value |
|---|---|---|
| 03 Readout correction | Temperature / pressure / relative humidity | 23.4 °C / 99.1 kPa / 45% |
| 04 Repeatability | Repeat readings | 1.482 1.495 1.471 1.488 1.479 |
| 04 Repeatability | Apply kρ to each reading | on |
| 04 Repeatability | Type B: calibration certificate (reference chamber) | 1.0% (relative standard uncertainty) |
Results
| Step | Value |
|---|---|
| Air-density correction kρ | 1.0336 |
| Corrected readings | 1.5317, 1.5452, 1.5204, 1.5379, 1.5286 |
| Mean x̄ / standard deviation s | 1.5328 / 0.00939 |
| Standard uncertainty of the mean u = s/√5 | 0.00420 (0.274%) |
| Degrees of freedom ν / coverage factor k | 4 / 2.776 |
| Type A alone: x̄ ± U | 1.5328 ± 0.012 |
| Combined relative standard uncertainty | 1.037% |
| Effective degrees of freedom νeff | 821 |
| Reported result | 1.5328 ± 0.031 (k = 1.96, about 95%) |
What the numbers say
The correction is larger than the uncertainty. kρ raises every reading by 3.4%, while the whole expanded uncertainty is 2.0%. A forgotten correction is therefore not a matter of a slightly wider interval; it moves the result outside it. Temperature and pressure alone would give 296.55/293.15 × 101.325/99.1 = 1.0343. The humidity term brings that down to 1.0336, because air at 45% relative humidity is slightly denser than the 65% reference.
The Type A coverage factor does not survive the combination. With five readings, t0.975, 4 = 2.776 is right for the repeatability on its own. Once the 1.0% certificate component, with its unlimited degrees of freedom, is added in quadrature, Welch–Satterthwaite gives νeff ≈ 821 and the coverage factor drops to 1.96. Multiplying the combined 1.037% by 2.776 instead would report 2.88% rather than 2.04%, an interval 41% too wide.
When the certificate is good, repeatability takes over. Changing the certificate entry to 0.2% gives νeff = 9.4 and k = 2.25, so the expanded uncertainty is 0.76%, no longer dominated by the reference. With only the first three readings and the 1.0% certificate, the Type A factor jumps to 4.303 while the combined k stays at 2.00. The cheapest way to tighten a budget depends on which component dominates, and the component table in panel D shows that at a glance.
A real budget has more than one Type B line. Positioning is a typical next entry: an uncertainty of 3 mm in a 1 m chamber distance is 0.6% in air kerma through the inverse-square law. Adding it with + Type B component raises the combined relative standard uncertainty from 1.037% to 1.198%, νeff to about 1463 and the expanded uncertainty to 0.036 (2.35%), with k still 1.96. Each new line is cheap to add and shows immediately whether it matters next to the certificate.
Round the report, not the arithmetic. The panel shows the value to five significant figures so that nothing is lost between steps. For a certificate, round U to two significant figures and the value to the same decimal place: 1.533 ± 0.031.
The formulas behind each row are on the Methods page, and the Guide lists the other inputs that change an answer.
Sources
- JCGM 100:2008, Guide to the expression of uncertainty in measurement: 4.2 (Type A), 5.1.2 (combination), G.3 (Student t), G.4 (Welch–Satterthwaite), 7.2.6 (number of digits).
- O. A. Alduchov, R. E. Eskridge, Improved Magnus form approximation of saturation vapor pressure, Journal of Applied Meteorology 35 (1996) 601–609.
- Reference conditions 20 °C, 101.325 kPa, 65% relative humidity as used by the workbench for vented chambers; implementation in air-density.ts.