Calculation of Expanded Uncertainties for CO2

Last update: February 2017 (Brad Hall)

The general method for estimating the expanded (total) uncertainty of the X2007 CO2 mole fraction scale is outlined here.


ppm = µmol mol-1
primary = CO2 primary standard (modified natural air)
standard uncertainty = approx. 1-sigma, (68% confidence level)
expanded uncertainty = approx. 2-sigma, (95% confidence level)
X = mole fraction in dry air
Φ = volume ratio (large volume relative to small volume)
P = pressure
T = temperature
uξ = standard uncertainty estimated for the quantity ξ (T, P, Φ, etc)

Measurement Equation:

The mole fraction of CO2 in a sample of dry air is calculated as:

where R is the gas constant, β is the second Virial coefficient, Φ is the volume ratio, and XN2O is the mole fraction of N2O.

Estimating the total uncertainty involves the following steps:

  1. Estimate uT and uP applicable to a volume ratio experiment. a. Here we use the “metRology” software developed by NIST, which calculates partial derivatives and performs the error propagation for a given mean value and standard uncertainty of each variable in a measurement equation. The software accounts for correlations among variables, which were determined separately following JCGM (2008). An alternate software tool, available at, gives similar results.
  2. Calculate uΦ.
  3. Estimate the uncertainties of pressure (uP) and temperature (uT) measurement during a manometer run.
  4. Calculate uCO2 for a single manometer run and then include typical repeatability of manometer runs of the same primary standard. This gives the standard uncertainty for analysis of a primary standard with mole fraction XCO2.
  5. Combine results from 4 with an estimate of the scale transfer uncertainty (reproducibility).

Steps 1-2

Uncertainties associated with pressure measurement are estimated from pressure calibrations performed using a piston gauge (primary pressure standard), manufacturer’s specifications, and variability of the pressure zero reading at vacuum.

Uncertainties on temperature measurement are estimated from calibration certificates obtained by accredited calibration facilities, and variability among different sensors placed at the same location inside a temperature-controlled oven. The repeatability associated with temperature measurement using a single probe is small and is ignored.

The volume ratio (VR) is calculated from successive expansions of air in four steps, bridging the difference between the large and small volumes.

Φ = r1*r2*r3*r4 - r1*r2*r3 + r1     (2)

Where r1, r2, r3, and r4 are intermediate volume ratios determined at each expansion step. Typical central values (mean) and standard uncertainties are shown in Table 1.

Table 1: Parameters of a typical series of volume expansions, P1 ==> P2, used to determine the volume ratio.
P1 (kPa) uP1 (kPa) P2 (kPa) uP2 (kPa) ri uri
80 0.0020 20 0.0015 4.15 0.00036
80 0.0020 10 0.0012 7.95 0.00109
80 0.0020 18 0.0015 4.53 0.00043
80 0.0020 12 0.0012 6.85 0.00074

Using the “metRology” software we calculate the standard uncertainty on the volume ratio to be 0.206 for a volume ratio of 880.10, or ~0.023%.

Step 3

For a manometer run, we estimate uT_air = 0.009 deg C, and uT_CO2 = 0.015 deg C. uT_CO2 is larger than uT_air because the oven temperature is disturbed upon freezing the CO2 sample at liq-N2 temperature, and takes some time to equilibrate. T_air is determined from the average of 3 PRTs and one thermistor, while T_CO2 is determined from the average of 2 PRTs. For pressure, we estimate uP_air = 0.0025 kPa and uP_CO2 = 0.0022 kPa. There is a slight difference here because P_air is determined near the upper range of the pressure calibration, while P_CO2 is closer to midrange. These uP are likely overestimated based on current run procedures, but we use them here because the CO2 scale is based on several years of data, so we want to estimate “typical” values from the past decade for this calculation. We also include uncertainty in the second Virial coefficient, β, taken from Zhao and Tans (1997).

Step 4

Using central values and standard uncertainties for T, P, VR, and β for typical measurement conditions (again, using the “metRology” software), we estimate the standard uncertainty for a single manometer run as a function of CO2 mole fraction. We then combine this estimate with the repeatability of manometer results (8 episodes over 20 years), and the analytical reproducibility.

Table 2: Uncertainty Budget at nominal 400 µmol/mol
Variable Typical value Standard uncertainty unit Contribution to uncertainty Relative contribution to uncertainty
Pair 83 0.0025 kPa 1.198E-08 7%
PCO2 30 0.0022 kPa 3.018E-08 17%
Tair 310 0.009 K 1.152E-08 6%
TCO2 310 0.015 K 1.920E-08 11%
Φ (vol. ratio) 880.10 0.206 dimensionless 9.400E-08 53%
Βair -5.87 0.2 cm3/mol 8.940E-14 0%
ΒCO2 -112.8 0.2 cm3/mol 2.560E-13 0%
XN2O 0.325 0.0005 nmol/mol 2.441E-12 0%
repeatability, N=8   0.035 μmol/mol 7.813E-09 5%
analytical reproducibility*   0.01 µmol/mol 1.2500E-09 <1%
* based on less than one year of data

Step 5

Combining uncertainties from Table 2, we obtain an estimate for the standard uncertainty. Repeating this exercise at different mole fraction we obtain the results shown in Table 3, and then multiply by coverage factor k=2 to obtain the expanded uncertainty (Figure 1).

Result: uCO2 = 0.22 ppm (0.055%), at 400 ppm CO2 (coverage factor k=2, ~95% confidence level)

Table 3: Standard uncertainty estimates including scale transfer.
CO2 (ppm) Standard uncertainty (ppm)
250 0.079
300 0.090
350 0.100
400 0.110
450 0.120
500 0.132
550 0.142

Figure 1: Expanded uncertainty estimate for CO2 (X2007 scale), expressed in mol mol-1 (black line) and in percent (red line). A simplified expression for the expanded uncertainty as a function of mole fraction is uCO2 = 0.000428*XCO2 + 0.05 (ppm).


  • JCGM 100:2008 Evaluation of Measurement Data – Guide to the Expression of Uncertainty in Measurement (ISO GUM 1995 with minor corrections), Joint Committee for Guides in Metrology (2008);
  • Tans et al (2015), “History of WMO CO2 X2007 scale: long-term reproducibility”, presented at the WMO/IAEA GGMT Meeting, Sept. 13-17, 2015; La Jolla, CA.
  • Zhao, C., P.P. Tans, and K.W. Thoning (1997), A high precision manometric system for absolute calibrations of CO2 in dry air, J. Geophys. Res. 102, 5885-5894.
  • Zhao, C., and P.P. Tans (2006), Estimating uncertainty of the WMO Mole Fraction Scale for carbon dioxide in air, J. Geophys. Res. 111, D08S09, doi: 10.1029/2005JD006003.