1. "Propagating" Degrees of Freedom

Good evening!

-Background
I'm a chemistry student who's received some elementary training in statistics. At the moment, I need to take experimental data, perform a series of calculations with it, and then provide a confidence interval for the final result using the t-distribution. I just have a question regarding how many degrees of freedom this final result will have. I've tried in vain to find data online about carrying degrees of freedom through a series of calculations.

My dataset consists of three groups of measured values.
Say n1 = 10, n2 = 8, n3 = 7.
I have to do a linear regression on each of these. I understand that this removes two degrees of freedom from each set, so I'm left with slopes m1, m2, and m3 which have 8, 6, and 5 df respectively.

Now, if I calculate the average of these slopes, how many degrees of freedom does it have? Two (3-1), or 18 ((8+6+5)-1)?

Afterwards, I do a series of calculations with this average involving a bunch of physical constants. In one case, I square the result. How can I carry the degrees of freedom all the way to my final answer? Is there a table of formulas for propagating degrees of freedom through a variety of operations, as there is for propagation of error? My searches have yielded little on this, but I'm not afraid to read a bit, so if you could direct me even to a link somewhere I would be very grateful. I've read the Wikipedia page on degrees of freedom and explanations on a few other sites, but I don't understand enough about some of the models to know if they apply to my case.

Thanks much!

2. Re: "Propagating" Degrees of Freedom

What is the purpose of averaging the slopes?

3. Re: "Propagating" Degrees of Freedom

I am recreating the classical experiment of measuring the speed of sound in a gas and using that to calculate its heat capacity ratio. I begin by using a Kundt's Tube to determine the distance between each half-wavelength of a sound wave at a certain frequency in the gas. The displacements are my raw data. Plotting displacement vs. number of half-wavelengths counted, the slope is the wavelength λ of the sound wave. I did this at three frequencies for each gas, and dividing the slope by the frequency according to λ/v = s gives me the speed of sound in that gas. It was actually this speed, and not the slopes themselves, that I averaged. As to the purpose of averaging the speeds, [because I was told to - I'm not sure what my professor's motivations are]. While I've been told that averaging needlessly reduces precision, I'm guessing it also helps to reduce systematic error in this experiment if say, my signal generator is not as exact at producing a tone of one frequency over another (and this is the case: the error on its output frequency is +-2%); and since the speed of sound should be the same no matter the frequency, using multiple frequencies means more sources of data and improved accuracy.

My question of how to propagate df through these calculations remains the same, I think. Does s have the same amount of degrees of freedom as λ because λ is only being divided by a constant? Does the average of the speeds have 2 degrees of freedom, because there are three speeds in it, or does it have 18 from the individual measurements?

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