# Error calculations for an experiment involving the collection rate of a certain material onto a wire

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CloudNine
Hi all,
I'm having a bit hard time performing error calculations on one of the lab results I got.
The lab dealt with finding collection rate of a certain material onto a wire. While inside a vial, the wire was soaked in the material for a specific period of time. Then the wire was taken out of the vial and both the wire and the vial with the remaining contents were measured.

Note that it is given that the collection rate is governed by Poisson distribution.

The collection rate is simply given by: CR=W/(W+R)
where: CR = collection rate; W = measuring the material content which stuck on the wire; R = measuring the material left in the residue in the vial

I have an excel sheet with a couple of such measurements, and now I would like to calculate the relative error in this experiment.

From what I know, I'm dealing with a reciprocal error calculation:

(*) Error = sqrt((1/W+R)^2*sW^2) + (-W/R^2)^2*s(W+R)^2)

My questions are:

1. Does sW and s(W+R) simply equal to sqrt(W) and sqrt(W+R), respectively?
2. Since one of the parameters is a sum of 2 values - W+R, should I calculate its error by using sum-error formula: sqrt(sW^2+sR^2) rather than treating this number as a united parameter as I wrote on point 1.? But what should I plug instead of sR? I feel like I'm stuck in a recursive loop.
Also, would it require me to modify the overall error formula I used (*) in any way?

Thanks!

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What is the nature of the material that you are collecting? How are quantifying the amount of material on the wire or in the vial?

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A few questions
1. Why is CR called a "rate"? To me a rate is "stuff" per time. Do you mean collection fraction ??
2. What is sR (nomenclature) and where does it come from ?

Why is CR called a "rate"? To me a rate is "stuff" per time.
Rate could also be an occurrence of stuff per unit of length, volume, gram, etc as in bacteria per cc or cars per mile depending on the sampling method.

CloudNine
What is the nature of the material that you are collecting? How are quantifying the amount of material on the wire or in the vial?
Didn't want to add complexity to the story, but maybe it is inevitable :)
I'm collecting a radioactive material on top of a wire, and so I measure the CPMs (with a gamma detector) on both the wire and the residue to see the amount of activity that was attached to the wire.

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Rate could also be an occurrence of stuff per unit of length, volume, gram, etc as in bacteria per cc or cars per mile depending on the sampling method.
Sure but the definition is a dimensionless ratio and so I am mystified.

CloudNine
A few questions
1. Why is CR called a "rate"? To me a rate is "stuff" per time. Do you mean collection fraction ??
2. What is sR (nomenclature) and where does it come from ?
You are correct, indeed collection fraction would be the correct term.
sR is the the error in R (as in ∆x in the original formula notation)

hutchphd
Didn't want to add complexity to the story, but maybe it is inevitable :)
I'm collecting a radioactive material on top of a wire, and so I measure the CPMs (with a gamma detector) on both the wire and the residue to see the amount of activity that was attached to the wire.

This was necessary as the Poisson distribution is for the occurrence of events that your OP left open for speculation. So the uncertainty in your measurement is indeed the square root of the number of counts in a given time interval.

CloudNine
This was necessary as the Poisson distribution is for the occurrence of events that your OP left open for speculation. So the uncertainty in your measurement is indeed the square root of the number of counts in a given time interval.
Both of my questions remain though, I kinda knew that I should use the Poison theorem for the calculations :) How do I deal with the "complex" term W+R in the error propagation?

Gezstarski
Care is needed here in several respects
1) For Poisson Statsistics to be used it is indeed important that W and R are not rates, but counts in the same interval of time. In that case the standard errors on W and on R+W are indeed Sqrt(R) and Sqrt(R+W) ie the sqrt of the counts in each case
You could equally get to s(R+W) by saying it is Sqrt (s(R)^2 + s(W)^2) = Sqrt ( R + W)
2) The formula for combining errors as the sqrt of sum of squares assumes that the errors are independent. The errors on R and on R+W are not independent. You need to work through the effect on CR of an error in R and and of an error on W and then combine THOSE as the sqrt of the sum of the errors

Yes, the quantities W and R are correlated. The propagation of errors formula includes a cross term and needs inclusion and consideration in your error formula. Using your notation this term is

+2·(sWR)2·(∂CR/∂W)·(∂CR/∂R)

With sWR being the covariance of W and R.

I believe that you could then write this term as ≈ 2·(ssR)·(-W / (W+R)3)

However, if W<<R then this is negligible.

Gezstarski
Its W and (W+R) that are correlated

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Much of this is not reasonable to me. The question is not whether the quantities are correlated but whether the errors in the reported measurements are. These measurement errors are very likely independent.
And why do you need to measure W+R repeatedly? Isn't W+R a constant = R( at t=0 when W=0) .
Ignore me if I don't understand your experiment...

Its W and (W+R) that are correlated

Let W+R = X so CR =W/X

Error CR = Sqrt[ s(W)2·(∂CR/∂W)2+s(X)2·(∂CR/∂X)2+2·s(W)·s(X)·(∂CR/∂W)·(∂CR/∂X)]

And why do you need to measure W+R repeatedly? Isn't W+R a constant = R( at t=0 when W=0) .
There is only one measurement.

Gezstarski
I think Gleem's formula is correct, but its messy to expand. Its easier to consider the inverse ratio D = 1/CR = (W+R)/W = 1+(R/W) and then use s(CR)/R = s(D)/D.

CloudNine
Let W+R = X so CR =W/X

Error CR = Sqrt[ s(W)2·(∂CR/∂W)2+s(X)2·(∂CR/∂X)2+2·s(W)·s(X)·(∂CR/∂W)·(∂CR/∂X)]

There is only one measurement.

just so I make sure I didn't get lost...:
s(W) = sqrt(W)
s(W)2 = W
Correct?

gleem
Of course, the random errors from decay are not the only errors to consider.

CloudNine
Of course, the random errors from decay are not the only errors to consider.
Not sure to what you are referring in this statement? :P
Just wanted to make sure that s(W)2 simply equals to w, because I'm not sure if it equals to [s(W)]^2 or not.

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Sorry... still lost. What is being measured and how?

CloudNine
Sorry... still lost. What is being measured and how?
CPM is measured - the gamma readings of the sample.
So one measure for the wire -> W
second measure for the vial with the remaining material -> R

hutchphd
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And you make a series of measurements for different "wire in fluid" durations?

Not sure to what you are referring in this statement? :P
General setup, instrumental or procedural errors. They may be insignificant it or negligible. I do not know your setup and procedure so I cannot be specific but you are counting the wire and vial in different geometries that can affect your relative counts. Your count rates could be significantly different resulting in different dead times, The backgrounds could be different from the wire to the vial.

hutchphd
@CloudNine One thing occurred to me. When you withdraw the wire from the solution I would think some of the original solution adheres to the wire and is not part of the purpose that you are looking for activity on it. One would need to correct for this or remove it which undoubtedly has some error associated with it.

Gezstarski
I think Gleem's formula is correct, but its messy to expand. Its easier to consider the inverse ratio D = 1/CR = (W+R)/W = 1+(R/W) and then use s(CR)/R = s(D)/D.

2) I am a bit worried about the statement "CPM is being measured" If "CPM" is Counts per minute then you can't use
Error= SQRT(value)
unless the measure is over exactly 1 minute. Fir thius tio be valid "value" must be a count, not a count rate.
Strictly there is another caveat - value must not be too small, because strictly it should be SQRT(true value). If you only get 1 or 2 counts, or even none, the true expectation value is pretty uncertain.

3) Following up my comment

I think Gleem's formula is correct, but its messy to expand. Its easier to consider the inverse ratio D = 1/CR = (W+R)/W = 1+(R/W) and then use s(CR)/R = s(D)/D.

Here is the full calculation ( forgive my change of nomenclature, but I think its clearer)

If you use rate then you can use this.

S..d of rate = ± n1/2/t where n is the number of counts in the counting period t.

CloudNine

2) I am a bit worried about the statement "CPM is being measured" If "CPM" is Counts per minute then you can't use
Error= SQRT(value)
unless the measure is over exactly 1 minute. Fir thius tio be valid "value" must be a count, not a count rate.
Strictly there is another caveat - value must not be too small, because strictly it should be SQRT(true value). If you only get 1 or 2 counts, or even none, the true expectation value is pretty uncertain.

3) Following up my comment

Here is the full calculation ( forgive my change of nomenclature, but I think its clearer)
View attachment 303087
Thank you so much! Everything is well explained :)
Can you just please explain why

Mentor
Why is CR called a "rate"? To me a rate is "stuff" per time.
Not necessarily. Rates of change of "stuff" per unit of time are sometimes called time rates of change. A derivative is the rate of change of one quantity with respect to some other quantity. Calculus textbooks usually have a section called "related rates," with examples concerning the rate of change of area relative to radius, or volume relative to height, and so on.

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Sure but the definition is a dimensionless ratio and so I am mystified.

Gezstarski
The interest rate on a bank account is a dimensionless number, too.

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The interest rate on a bank account is a dimensionless number, too.
No it is not. The usual (albeit implied) value is %/year.

Gezstarski
I accept - not a good example. But there are various tax 'rates' that are dimensionless.