Measurement uncertainty

In summary: But if it is, then it should be possible to estimate the standard deviation of D in terms of the estimated standard deviations for t and h.In summary, the Magnus formula is used to calculate the saturated water-vapour pressure Ew at temperature t. The average of the 10 repeated measurements of air temperature is 22.27 ˚C with a standard deviation of 0.09 ˚C, while 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 a standard deviation of the mean of 0.10 %rh. The correction of the hygrometer is -1.
  • #1
Ronalds
3
0

Homework Statement


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%.

Please help at least somehow with this task which is hard to solve! I know that solution is too much to ask, and will appreaciate every help!

Homework 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))

The Attempt at a Solution

 
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  • #2
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.
 

1. What is measurement uncertainty?

Measurement uncertainty is the degree of doubt or variability that exists when determining the value of a quantity. It is a measure of the confidence we have in the result of a measurement and is affected by various factors such as the precision of the instrument, human error, and environmental conditions.

2. Why is it important to consider measurement uncertainty?

Measurement uncertainty is important because it provides a realistic representation of the accuracy of a measurement. It allows us to understand the potential errors and limitations in our measurements, which is crucial for making informed decisions and drawing valid conclusions from scientific data.

3. How is measurement uncertainty calculated?

Measurement uncertainty is typically calculated using statistical methods such as standard deviation or confidence intervals. It involves analyzing repeated measurements and estimating the range of values within which the true value of the quantity is likely to lie.

4. What are the sources of measurement uncertainty?

Measurement uncertainty can arise from various sources, including instrumental factors, such as calibration and accuracy of the equipment, environmental conditions, such as temperature and humidity, and human factors, such as technique and perception. Other sources include natural variations and random errors in the measurement process.

5. How can measurement uncertainty be reduced?

Measurement uncertainty can be reduced by using precise and calibrated instruments, following standardized measurement procedures, and minimizing environmental and human factors. Additionally, conducting multiple measurements and using statistical analysis can help to reduce uncertainty and improve the accuracy of the measurement.

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