# Uncertainty of measurement: water vapour transmission of materials

The rate at which water vapour is transmitted through a sheet material is important, for example, if the sheet is to be used to damp-proof a building. The documentary standards ASTM E96 and SANS 952-1 describe methods of evaluating water vapour transmission (WVT), using a desiccant or water to establish a humidity gradient across the sheet. The following article will briefly describe the method, and then focus on estimation of the uncertainty of the calculated WVT value.

The Desiccant Method [E96 clause 4.1] involves sealing the specimen to be tested over the mouth of an impermeable test dish, with desiccant (anhydrous calcium chloride) inside. This dish is placed inside a test chamber or enclosure, where the temperature and humidity are controlled at (38 ± 1) °C and (95 ± 5) %rh [952-1 clause 6.11.1.5], for example. The dish is briefly removed from the chamber and weighed, periodically, over a period of several days. The mass gain indicates the amount of water vapour moving through the specimen into the desiccant. Typically, three specimens are tested [E96 clause 9.1], with a fourth “dummy” (control) specimen mounted on a dish and weighed, but without any desiccant or water in the dish to maintain a humidity gradient [E96 clause 9.6]. The mass change of the dummy is subtracted from the change in each of the test specimens, to cancel out the effect of variations in temperature and air buoyancy [E96 clause 11.3]. (In practice, vigorous air circulation within the test chamber, required to keep the surrounding air uniform in temperature and humidity [E96 clause 6.2], causes the dummy to gain significant mass, too, because it was sealed at a lower ambient humidity than the 95 %rh maintained in the chamber.)

Measurements of specimen mass are performed (perhaps once every 24 hours), until approximately six consecutive changes in mass (corrected for variation in the dummy), plotted versus time, fall “on a reasonably straight line” [952-1 clause 6.11.4.2.3]. The slope of this line (equal to rate of mass change, in grams per hour) is converted to WVT in grams per square metre per 24 hours [952-1 clause 6.11.5], using the measured area of the mouth of each dish. The mean of the WVT values of the three specimens is reported to the customer.

Here is a data set, for use in our discussion below:

 Time Dummy Specimen 1 Specimen 2 Specimen 3 (h) m (g) dm (g) m (g) dm (g) dm_corr (g) m (g) dm (g) dm_corr (g) m (g) dm (g) dm_corr (g) 0 163.4055 0.0000 197.6220 0.0000 0.0000 200.0488 0.0000 0.0000 190.0098 0.0000 0.0000 24 163.4061 0.0006 197.6260 0.0040 0.0034 200.0545 0.0057 0.0051 190.0167 0.0069 0.0063 48 163.4112 0.0057 197.6346 0.0126 0.0069 200.0615 0.0127 0.0070 190.0238 0.0140 0.0083 72 163.4148 0.0093 197.6408 0.0188 0.0095 200.0685 0.0197 0.0104 190.0326 0.0228 0.0135 120 163.4165 0.0110 197.6462 0.0242 0.0132 200.0753 0.0265 0.0155 190.0392 0.0294 0.0184 144 163.4182 0.0127 197.6554 0.0334 0.0207 200.0847 0.0359 0.0232 190.0471 0.0373 0.0246 Fitted slope (g/h): 0.000130 0.000145 0.000158 Area (mm^2): 3117 3117 3117 WVT (g/m^2/24h): 1.00 1.12 1.21

The slopes above were fitted using the LINEST spreadsheet function. Using the formula =LINEST(measured_y_values, measured_x_values, intercept, stats), with intercept=TRUE (allowing a non-zero y-intercept) and stats=TRUE (displaying not just the fitted values of slope and intercept, but also four additional rows of statistical information), for specimen 1, the following output is obtained:

 0.000 130 0.000 11 0.000 013 0.001 09 0.963 0.001 59 103.10 4 0.000 26 0.000 01

The first row gives fitted slope and y-intercept, the second row standard errors (uncertainties at coverage factor k=1) of the fitted slope and intercept, the third row correlation coefficient R^2 and standard error in the y-values, the fourth row the Fisher F-statistic and the degrees of freedom (number of data points minus number of fitted parameters). We will discuss the use of some of these statistical measures below. (The fifth row is not of interest to this discussion.)

How may we estimate the uncertainty of measurement (UoM) of this WVT value? First, consider that the basic task is to determine the change in mass over the change in time, or $Latex formula$. The absolute accuracy of the balance used to weigh the dishes is not important, nor are the absolute times indicated by the watch: only the changes are of interest.

How can we estimate the accuracy of $Latex formula$? We could evaluate the sensitivity of the balance [OIML R 76-1 clause T.4.1] to small changes in a load around 200 g (in the above example). However, there is an easier way, which also takes into account any effects that vary randomly from one balance reading to the next. As we have six data points to determine only two unknowns (slope and y-intercept of the straight line), we have what is called an “over-determined system”, for which we find the “best fit” line by the method of least squares. This line typically does not pass through any of the data points, but is “as close as possible” to all (in other words, the best compromise to all available data). The differences between the measured data points and the fitted line give a quantitative estimate of random errors in the measured values. These differences are conveniently combined in the standard error of the fitted slope, namely, 0.000 013 g/h for specimen 1 above. How do we expand this standard (k=1) uncertainty, to reach a level of confidence of approximately 95%? For a very large data set, we would simply multiply u(k=1) by the coverage factor k=2. However, as our data set is small, we should enlarge the coverage factor somewhat, to compensate for our limited knowledge. The t-distribution of Student tells us what this k-value should be: the spreadsheet function =TINV(1-0.95, 4) gives the value of Student’s t-distribution for 4 degrees of freedom and 95% level of confidence, namely, 2.8. Multiplying u(k=1) = 0.000 013 g/h by 2.8, we obtain the expanded uncertainty 0.000 036 g/h. (This is what U(k=2) would have been, if we had a very large data set, or, in other words, almost infinite degreees of freedom.) So, the slope for specimen 1 is (0.000 130 ± 0.000 036) g/h. Or, as a relative uncertainty, $Latex formula$, or 27%. Note that this is significantly worse than the balance sensitivity of 1% of total mass gain (which would translate to 0.000 001 g/h) required in the test methods [E96 clause 6.3, 952-1 clause 6.11.4.2.2], indicating that balance sensitivity is, in this case, negligible compared to other factors causing “noise” in the mass readings. It highlights the importance of having more data points than unknowns and performing a least squares fit, as this uncertainty component would otherwise be grossly underestimated. It also shows how the standard error in the slope is far more useful than the correlation coefficient R^2, in quantifying uncertainties.

Now, how do we estimate the uncertainty in $Latex formula$? (In the previous paragraph, it seems we already obtained a “complete” uncertainty for the slope. However, the fitting procedure we used assumes no uncertainty in the x-values, only uncertainty in the y-values, so we had better look at the uncertainty in the time intervals, too.) The documentary standards require time intervals to be measured to an accuracy of 1% (for example, a 24 hour interval to an accuracy of 15 minutes), so, $Latex formula$ [E96 clause 11.3, 952-1 clause 6.11.4.2.2]. We can see that the relative uncertainty in $Latex formula$ is far smaller than that in the fitted slope, so we expect the uncertainty in slope to completely dominate the final, combined, uncertainty.

The formula for WVT is $Latex formula$. When parameters are combined by simple multiplication or division, the relative uncertainties may be added simply in quadrature. So, $Latex formula$. As $Latex formula$ is typically smaller than 0.01, it, like $Latex formula$, can be neglected, so that the final relative uncertainty is $Latex formula$, or 27%. In other words, U(k=2) = 1.00 g/m^2/24h * 0.27 = 0.27 g/m^2/24h.

The above is the UoM for specimen 1. Specimens 2 and 3 have similar uncertainties (0.29 g/m^2/24h and 0.25 g/m^2/24h, respectively). Now, what is the uncertainty of the mean of the three specimens? If the three uncertainties are similar in magnitude, the uncertainty of the mean is $Latex formula$. If the uncertainties are quite different, the weighted mean (giving more weight to those specimens with smaller uncertainties) could be used: $Latex formula$. The uncertainty of the weighted mean is $Latex formula$, which also works out to be 0.16 g/m^2/24h, in the above example.