Computing Errors when Data Sets are Given

[Wannabe Physicist]

Homework Statement

There are three relevant quantities, $L,m,n$. I have been given: $L = 7.056 \pm 0.005$ cm. I also have the data set for measurements of $m$ taken as function of $\Delta p$. I'd like to find the error in $n$

Relevant Equations

$n = 1+\displaystyle\frac{\lambda p m}{2L\Delta p}$

This is for the lab report I have to submit. $n$ is the refractive index. $L$ is the length of a gas chamber and $m$ is the number of fringes passed as the pressure in the gas chamber changes by $\Delta p$. We are already given the error in $L$. I performed the experiment and obtained 16 observations. So I have the data sets $m = \{m_i\}_{i=1}^{16}$ and $\Delta p = \{\Delta p_i\}_{i=1}^{16}$.

Now I am confused as to how to proceed to calculate the error in $n$. I should obviously consider the error in $L$. But how should I use the data sets to compute the error? Should the formula be
$$\frac{\delta n}{n} = \frac{\delta L}{L} + \frac{\sigma_m}{\bar{m}\cdot \sqrt{16}} + \frac{\sigma_{\Delta p}}{\bar{\Delta p}\cdot \sqrt{16}}$$
where $\bar{m}$ and $\sigma_m$ are the mean and standard deviation for $m$ respectively and notations are similar for $\Delta p$

A classmate of mine suggested using the least count of the pressure gauge (Pressure gauge was calibrated in Torr with least count = 10 Torr). But I don't know if this should be used.

 

[haruspex]

$\sigma_m$ and $\sigma_{\Delta p}$ are consequences of the choices made in conducting the experiment. Their ranges do not reflect any kind of uncertainty in the data.
It is unclear to me how m and ${\Delta p}$ were determined. I'll assume the pressure was changed until m changed then both recorded. That being so, there is no error in m.
The input from these data to the rest of the calculation consists of the slope of the straight line fit to ${\Delta p}$ plotted against m.
You know this must pass through the origin, but I am not sure offhand how one uses that... we can return to that if necessary.
So you need to find uncertainty in the slope given the data. Google for that.

 

[Wannabe Physicist]

The readings were taken by reducing the pressure by regular intervals of 10 torr and counting the number of fringes that moved outwards within that interval.

I have already plotted the graph of $m$ vs. $\Delta p$ and can reasonably say my error in counting $m$ should be $\pm 1$. I found this link which tells how to find the uncertainty of slope: http://spiff.rit.edu/classes/phys311/workshops/w2c/slope_uncert.html

With this help, the slope of my plot is $0.1 \pm 0.0135$ Torr$^{-1}$

So now I think the error formula should be
$$\frac{\delta n}{n} = \frac{\delta L}{L} + \frac{0.0135}{0.1}$$
Is this formula correct?

 

[haruspex]

The readings were taken by reducing the pressure by regular intervals of 10 torr and counting the number of fringes that moved outwards within that interval.

I have already plotted the graph of $m$ vs. $\Delta p$ and can reasonably say my error in counting $m$ should be $\pm 1$. I found this link which tells how to find the uncertainty of slope: http://spiff.rit.edu/classes/phys311/workshops/w2c/slope_uncert.html

With this help, the slope of my plot is $0.1 \pm 0.0135$ Torr$^{-1}$

So now I think the error formula should be
$$\frac{\delta n}{n} = \frac{\delta L}{L} + \frac{0.0135}{0.1}$$
Is this formula correct?

Looks reasonable to me.

 

[Wannabe Physicist]

Okay. Thanks a lot. One last question: We used an analog pressure gauge. So is the error in $\Delta p=$ least count of pressure gauge? Or should I take no error in $\Delta p$?

For my post #3 I assumed no error

 

[haruspex]

Okay. Thanks a lot. One last question: We used an analog pressure gauge. So is the error in $\Delta p=$ least count of pressure gauge? Or should I take no error in $\Delta p$?

For my post #3 I assumed no error

Are you saying the reading is to the nearest 10Torr? If so, set the error to 5.

 

[Wannabe Physicist]

Yes. Thanks a lot