Finding the mean value from multiple abs+-error data

[pinsky]

Hello!

I suppose the title didn't really describe it, but here is the case. I have some data about speed of sound in air which i got by measuring distance and wavelength (the Kundt's tube experiment).

So now i have three different results for speed which are all in the "Mean value" +- error form. How do i get one mean value and error? How do errors influence the outcome of the result?

There are

v₁=324.9+-13 [m/s]
v₂=347.76+-13 [m/s]
v₃=362.9+-7.24 [m/s]

so I'm looking for v_mean+- error.


Any advice is appreciated.

 

[D H]

You could use an error-weighted mean. However, there is an implicit assumption that the values are consistent with one another in the derivation of the equation for the error-weighted mean.

Do you have a consistent data set here?

 

[pinsky]

It seems smy knowledge of the subject isn't sufficient to answer your question. :p

I'll tell you what I've done, and perhaps xou can help me determine if the data sets are consistent.

First.

Ive measured frequencies and wavelength's three times for three different frequencies.
As a resule i got three pairs ov data.

$$f_1\pm \frac{\sigma_{f1}}{\sqrt{3}} \;\;\;\; \lambda_1/2 \pm \frac{\sigma_{\lambda_1/2 }}{\sqrt{3}}$$

Lambda is halved because that was the way it was measured. The number under the root is 3 because there were three measurements, to say it differently, there number of data from which I'm calculating the mean and error was three.

Second.

$$\lambda = 2 \cdot \lambda_1/2 \pm 2 \cdot \frac{\sigma_{\lambda_1/2 }}{\sqrt{3}}$$

Third.

$$v = \lambda \cdot f$$

$$\overline{v} = \overline{\lambda} \cdot \overline{f}$$

$$\left | \frac{\Delta v }{v} \right |= \sqrt{\left | \frac{\Delta \lambda }{\lambda} \right |^2+\left | \frac{\Delta f }{f} \right |^2}$$

So now, I have three values for the speed and am looking for their average value. Are the data sets consistent?

 

[pinsky]

I've just had a very simple idea for a solution.

I have three values

v₁=v_{1_mean}+- v_{1_error}
v₂=v_{2_mean}+- v_{2_error}
v₃=v_{3_mean}+- v_{3_error}

I need the average value of those three. How big of a mistake is it to just calculate the average values just as if they were mean values.

The values are just being summed, and scaled, there is no multiplication between them involved.
So what I'm basically doing is observing the average value of v as a function of v₁,v₂ and v₃

 

[D H]

You don't want to do that. You want an error-weighted mean. Have you googled that phrase? Your third measurement is much more precise than the other two. It should somehow get more weight in computing the average.

One standard approach is to compute

$$\bar x = \frac{\sum_i x_i/\sigma_i^2}{\sum_i 1/\sigma_i^2}$$

with those σ_i values being the uncertainty associated with the ith measurement. This is motivated by assuming that the measurements are independent but not identical distributions, and that the underlying distributions are close to normal.