# Measurement uncertainty

Tags:
1. Nov 27, 2014

### Ronalds

1. The problem statement, all variables and given/known data
The Magnus formula is used for calculating the saturated water-vapour pressure Ew at temperature t. The average of the 10 repeated measurements of air temperature is 22.27 ˚C with experimental standard deviation s(t)=0.09 ˚C. The permissible error of the thermometer is ±0.3 ˚C. The average of the 10 repeated measurements of relative humidity h is 38.9 %rh with corresponding standard deviation of the mean sA(h)=0.10 %rh. The correction of the hygrometer is -1.6 %rh and the corresponding expanded uncertainty is 1.2 %rh (k=2). The value of the molar mass of water-vapour Mw=0.018015 kg mol-1 and the value of the universal gas constant R=8.3145 J mol-1 K-1 . For calculations you can assume that Ew and t are fully correlated. Please calculate the absolute humidity Dw and estimate its expanded uncertainty at confidence level P=95%.

2. Relevant equations
In order to calculate the absolute humidity of air [g/m3] the following simplified equation can be used:

Dw = (1000 * Ew (t) * Mw * h) / (R * (t + 273.15))

3. The attempt at a solution

2. Nov 29, 2014

### Stephen Tashi

It's hard to advise you about this because the way "error" is treated varies among various fields of study and even among various people in those fields. It would help to see a simpler example where your instructor explained the method he prefers.

One way to treat such a problem is assume it is permissible to "linearize" the estimator. I don't know the physics in this problem. I gather that Dw (which I shall denote simply as "D") can be written as function of some variables so expanding D(t,h,...) about (t0,h0...) in terms of partial derivatives gives the approximation
D(t,h,..) = (t-t0) ((partial D/ parital t) evaluated at (t0,h0,..))
+ (h-h0)( (partial D/partial h) evaluated at (t0,h0,...))
+ ...

A typical approach is to take (t0,...) as the measured sample means. (Apply correction if there is a known bias - such as the - 1.6% for h) This means that we treat all the partial derivatives as constants.

The approximation thus as the form D(t,h,..) = (t-t0) c1 + (h-h0) c2 + .... where the c's are constants.
If we treat D(t,h,...) as a random variable then it is expressed as a sum of random variables. You can calculate the variance of D(t,h,..) if you assume it is the sum of independent random varaibles. Var (D) = Var(t-t0) (c1)^2 + Var(h-h0)(c2)^2 + ...

If you approximate the variances on the right hand side of the equation by the squares of the sample standard deviations, you can estimate a variance and standard deviation for D.

As I said, I don't know if this approach is what your instructor expects.