# Uncertainty of Derivatives

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1. Feb 28, 2016

### Jonathan Densil

1. The problem statement, all variables and given/known data
I know this is more of a physics question, but I tried there and wasn't successful.
I have done a physics experiment measuring the weight as a function time of the discharge of water from a cylindrical bottle with a pinhole at the bottom. What I ultimately want to get at is the weight flow rate as a function of time. I already have the equation describing this scenario, but I don't want to get a theoretical uncertainty, I want an experimental uncertainty for weight flow rate. To get weight flow rate, I took the numerical derivation of the weight using Vernier's LoggerPro software. The constant absolute uncertainty of the weight is $\pm 0.01$ N for all data points. Since I am taking the derivative of the weight values to get weight flow rate, what would be the uncertainty of the weight flow rate. If you need any more data or information, feel free to ask. I have around 2000 data points for 5 trials

2. Relevant equations
If the uncertainty of the weight is $\pm 0.01$ N, what is the uncertainty of the weight flow rate (derivative of the weight as a function of time)?

3. The attempt at a solution
When I looked this up on the web, almost all of the solutions involved some next level math that I didn't understand. I have been teaching myself calculus (limits, derivatives, integrals) on my own time since I am in Grade 11 (I only take calculus next year) Soooo... I don't know where to start.

2. Feb 28, 2016

### Buzz Bloom

Hi Jonathan:

I do not know enough to advise you how to complete the calculation, but here is where I would start.

Imagine calculating an estimate of the derivative at each data point. For each but the endpoints, find a quadratic that passes through the point and its neighbors. You can then have an expression for the value of the derivative for each quadratic. Do you know how to calculate the error range of the quadratic derivatives based on the error range of the data points?

Hope this helps.

Regards,
Buzz

3. Feb 28, 2016

### Jonathan Densil

I already have the derivatives at each point, I'm just not sure how to calculate the uncertainty for each derived point.

Kind regards,
Jonathan

4. Feb 28, 2016

### Buzz Bloom

Hi Jonathan:

I am not sure I understand the sequence of steps you used to get where you are. This is what I guess you did.

1. Collect measured data points givings pairs of values: weight and time.
2. Calculate a least mean squared (LMS) fit for some form of function w(t) with one or more coefficients to be determined by the LMS calculation. If this is correct, what is the form of the w(t) function?
3. You determined the error rate +/-0.01 N from the LMS fit.
4. You calculated the derivative w'(t) = d/dt w(t).

If this is correct, what you are missing is the equivalent of the raw measured values for w'(t). If you had those you could calculate the standard deviation of the differences between these raw measured values and w'(t). That would be the +/- error range of the derivative. One way to approximate equivalent measured values for the derivative data points is to use a quadratic fit as I described in post #3. There may be some better way if doing that than by using quadratics, but I think that the quadratics would give you a pretty good approximation.

Regards,
Buzz

5. Feb 28, 2016

### Jonathan Densil

The steps that I took:
1. Using the Vernier Dual Range Force Sensor, I measured the weight of the bottle as the water was flowing out of the hole at the bottom
2. I organized the data into a table of time on the left column and weight on the right column in Vernier LoggerPro software
3. Using the software, took a numerical derivation (calculated using its own means) of the weight as a function of time to get weight flow rate.
4. According to the Vernier Dual Range Force Sensor, the uncertainty given is 0.01 N.
I don't know what the uncertainty of the weight flow rate after LoggerPro took the derivation of the data points.

6. Feb 28, 2016

### Buzz Bloom

Hi Jonathan:

My guess is that the 0.01 N value is the standard deviation of an individual measurement using the DRFS device. Since I do not know the Vernier LoggerPro software, I am not sure what it is doing to calculate the derivatives. Is there a way for you to find out what the software does? If not, I suggest using a spreadsheet and the quadratic method to calculate the derivative at each time value and then the mean and standard deviation (SD) of the differences between those values and the software values. If the mean is very small compared with the SD, I would use +/- SD as an estimate for the error range.

BTW: If you average the weight measurements from the five trials, the resulting weight error range would be 0.01 divided by the square root of 5.

Good luck.

Regards,
Buzz

7. Feb 28, 2016

### Jonathan Densil

That is all of my trials data. I'm not aware of what the quadratic method is, sorry, can you please explain?