# Total error in a measurement.

1. ### Topher925

I have a question that has been bugging me lately. How is it that you determine the total error of a measurement?

For example, if we are trying to measure the flow rate of water coming out of a hose. We let the water flowing through the hose fill a graduated cylinder and measure the time it takes to do it. So we would have:

Flow rate = Volume / Time

However lets say that we need to know the error of this measurement. Would we say that the total error is:

Error = Ev*dQ/dv + Et*dQ/dt (d's are partial derivatives)

Where,
Q = function for flow rate
Ev = max error from volume measurement
Et = max error of time measurement
Error = total error

So the formula would ultimately be:

Error = Et*-V/t^2 + Ev*1/t

Would this be correct? Haven't done this in a while and its just not making sense to me?

Last edited: Sep 8, 2008
2. ### gmax137

911
I think you take the *partial* derivatives of the function wrt each variable, and multiply each pd by the error, then sum them up.

So in your example where F = V/t

you have err = ev d(V/t)/dV + et d(V/t)/dt (where these d's are partials)

You have to be careful to keep track of the units, sometimes errors are given as percent of scale and sometimes in absolute units.

3. ### Topher925

Thanks for the reply. Thats what I meant to say, I should probably correct that. I always get confused about this formula for some reason it just doesn't make much "physical" sense to me.

4. ### gmax137

911
The physical meaning is this - the partial derivative of f(x,y) wrt x is how much f changes for a given change in x. Now consider ex, the error in x, as that "given change in x". Then the change in f for the change in x is ex times partial of f wrt x.

5. ### Andy Resnick

5,895
John Taylor has written an excellent book "An Introduction to Error Analysis", and it's worth reading.

Under most conditions- errors in one quantity (say cylinder radius) are independent of errors in another quantity (say time to fill)- then the different errors add in quadrature. Another importnat assumption is that repeated measurements form a Gaussian distribution. Then, given a function of several variables q=F(x,y,v,...t) then:

$$\delta q = \sqrt{(\frac{\partial q}{\partial x} \delta x)^{2}+(\frac{\partial q}{\partial y} \delta y)^{2}+(\frac{\partial q}{\partial v} \delta v)^{2}+...(\frac{\partial q}{\partial t} \delta t)^{2}}$$

or variants thereof. The formula your wrote is also true, but it's an extremum value.

6. ### gmax137

911
sorry for any confusion - you are quite right about summing inquadrature (a.k.a, root-sum-square or RSS). This is probably where the OP has trouble with the "physical meaning." I remember when I was trying to get to understand the meaning of the RSS formula, I found Taylor's book and it helped me out. Now that was a few years ago and I can't remember how he explained it...And my copy of the book is not at hand...

7. ### f95toli

2,414
Error estimation is very tricky and there are often no single "correct" estimate. Nowadays Monte Carlo calculations are used in more advanced applications but even then there are problems; mainly related to how we interpret probability (i.e. what does the limits actually mean).
From what I understand even GUM* is a bit messy in that regard, mixing Baysian and functionalistic approaches.

*GUM=Guide for measurement uncertainty, a publication from ISO, IEC and a few other standardization organizations.

8. ### Nick89

550
I believe your equation for the error is correct, apart from absolute values!

I think it's supposed to be:

$$dQ = \left| \frac{\partial Q}{\partial V} \right| \, dV + \left| \frac{\partial Q}{\partial t} \right| \, dt = \frac{1}{t} \, dV + \frac{V}{t^2} \, dt$$
(Assuming t and V are both always positive as would be the obvious case with time and volume)

Note the + V/t^2..! It doesn't make sense to have your error become less (or even negative!) if some partial derivative happens to be negative...

9. ### Topher925

Thanks for all the replies. I am now a little more knowledgeable about error analysis. I'll try to get to that book someday but right now I just don't have the time due to other books that I have to read.

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