Measurement uncertainty of average value

[munja100]

This is my first post, so I would like to say hi to everybody.

 

This is not exactly homework, but I thought that this was the most appropriate sub-forum... I have problems with measurement uncertainty propagation, so I was wondering how other people cope with it. This is one problem that I couldn't solve so far.

 

We perform radon measurements, and each lasts 3 days. Let's denote the result of radon concentration with x_i, and associated uncertainty with u_xi. Now we need to calculate average monthly concentration, which is trivial, but also associated uncertainty, which proves to be almost impossible, for me at least. Obviously, standard deviation and standard error are inadequate, because we do not measure the same thing 10 times, but 10 different things. For example, if we can get results 100±1, 200±1, 300±2, the uncertainty of average value would be very low, because our measurements are very precise (less than 1%). However, standard deviation would be very large and is totally inadequate for this problem. The same is true if you have results like 30±8, 29±7, 31±6. Uncertainty of average is obviously much higher than standard deviation would indicate.

 

So, another solution is to use the formula for propagation of uncertainty, u²_f = Ʃ(±δf/δx_i)²u_xi². In our case, f is function for average, so f = 1/nƩx_i, so we get u²_f = 1/n²Ʃu_xi², which looks nice. However, imagine that all u_xi are the same values, or nearly the same. In that case, u²_f =1/n² * nu_xi², or u_f=u_xi/√n. Sooo, if we find the average of infinite number of radon concentration measurements, we have zero uncertainty. But it is not true, because there are some "systematic" errors associated with the same measurements and some of them are not random, i.e. they influence each measurement the same way. So we need also to take into account the correlation between measurement uncertainties, which seems very complicated to me. Each u_xi has at least 7 inputs and all would need to be investigated.

 

I asked around, but it seems that nobody in my surroundings has given it any thought. How do you deal with similar situation? Is there an easier way to calculate this, or do you use some approximation, or do you actually estimate all the covariances? Or do you somehow split u_xi² in two parts, one "systematic" and other random and then apply formula u²_f = 1/n²Ʃu_xi² for random part and add systematic effects to that?

 

 

[imatfaal]

This is out of my zone completely - but I think you need to read up on quantification and relationship between the two types of uncertainty

 

http://physics.nist.gov/cuu/Uncertainty/typea.html

http://physics.nist.gov/cuu/Uncertainty/typeb.html

 

I hope I haven't misunderstood

[studiot]

 

 

 

 

We perform radon measurements, and each lasts 3 days. Let's denote the result of radon concentration with x_i, and associated uncertainty with u_xi. Now we need to calculate average monthly concentration, which is trivial, but also associated uncertainty, which proves to be almost impossible, for me at least. Obviously, standard deviation and standard error are inadequate, because we do not measure the same thing 10 times, but 10 different things. For example, if we can get results 100±1, 200±1, 300±2, the uncertainty of average value would be very low, because our measurements are very precise (less than 1%). However, standard deviation would be very large and is totally inadequate for this problem. The same is true if you have results like 30±8, 29±7, 31±6. Uncertainty of average is obviously much higher than standard deviation would indicate.

 

You seem to have a good grasp of mathematical notation, but It is not clear to me what you are measuring.

 

Is the 100; 200; 300 a sequence of cumulative measurements of one location,

or individual estimates of small parts of one location

or what?

 

Statistical methods for treating cumulative frequency distributions are different from point value distributions.

 

You also say somerhing about measuring different things (underlined in your passage)

Can you explain this further?

 

I think once your methodology is sorted out, the appropriate techniques can be determined and applied.

 

[munja100]

For example, 100 is radon concentration between day 1 and day 3 of the measurement, 200 for days 4-6, 300 for days 7-9. And then you need average concentration and associated uncertainty. So not cumulative, just an unintentionally misleading example

 

Regarding measuring different things - I do not measure the same concentration 3 times, I'm measuring 3 different concentrations and I need average of that.

[Klaynos]

There's a good book on errors. An introduction to error analysis by John R Taylor.

 

Is your systematic error varying each time? If so, is the distribution of the systematic errors a normal distribution around 0?

 

If the answer to either of those two questions is no then error propagation is (probably) not the tool you're looking for.

 

Probably because I'd need to understand a lot more about what you're measuring and trying to achieve. Really I'd recommend anyone studying or trying to do science to read the book.

[studiot]

Hmm, let me see if I understand you.

 

 

For example, 100 is radon concentration between day 1 and day 3 of the measurement, 200 for days 4-6, 300 for days 7-9. And then you need average concentration and associated uncertainty. So not cumulative, just an unintentionally misleading example

Regarding measuring different things - I do not measure the same concentration 3 times, I'm measuring 3 different concentrations and I need average of that.

 

Thank you for the further details.

This may or may not have meaning.

 

So are you are measuring

 

1) Radon buildup in a 'radon proof' sampling vessel?

 

That is on days 1-3 100 units collect in the jar, on days 4-6 a further hundred units collect and on days 7-9 a further hundred units collect.

 

or

 

2) An increasing rate of radon escape

 

So that on days 1-3, the sampling jar collects 100 units, on days 4-6 a fresh sampling jar collects 200 units and on days 7-9 another fresh jar collects 300 units

 

Either way does an average have any real meaning?

 

If you have a bunch of numbers, you can always calculate an 'average', but this number does not always have any real meaning.

 

For instance, say I go into a hardware shop and buy 4 nails, 50mm, 75mm, 100mm, 150mm

 

Is the average nail length of any use?

 

But again say I buy 4 six inch nails of measured length 150mm, 149mm, 151mm, 150mm.

 

What does this average tell us?

 

An average is one measure of a single number used to represent a bunch of figures.

There are other measures that may be more appropriate.

Some might be the median, (half the values fall above and half below) or the mode (the peak value in a frequency curve)

 

But again it may be appropriate to report intervals rather than single values.

This is often the case, especially when readings or measurements are sparse and / you have no idea of the true population.

 

Then we come to the subject of errors and 'error propagation', which I don't think you are using appropriately from what you have so far described.

 

Are you actually asking

 

If my measurements are 100+- 3 ; 200+-5; 300+-8

 

Is the average 200 and if so how do I combine the tolerances to obtain an average tolerance?

 

(This is not error propagation by the way)

[munja100]

@Klaynos

 

Thank you, I will check the book

 

@studiot

 

Maybe I can try with another example. Three different people pour water in a tank from 5 different vessels. First one poured 100±7 liters, second 50±10 liters, third 150±20 liters, and they estimated uncertainty in the same way. How much did they pour in total? 300. What is the uncertainty of total? What if there is a correlated systematic effect in their measurement uncertainty, but also a random effect? (Also what is the average and what is its uncertainty?)

 

So average and total have meanings in this case.