Finding the mean value from multiple abs+-error data

In summary, the data sets are consistent, but you need to use an error-weighted mean to get the average value.
  • #1
pinsky
96
0
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 wavelenght (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 vmean+- error.


Any advice is appreciated.
 
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  • #2
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?
 
  • #3
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 wavelenght's three times for three different frequencies.
As a resule i got three pairs ov data.

[tex]f_1\pm \frac{\sigma_{f1}}{\sqrt{3}} \;\;\;\; \lambda_1/2 \pm \frac{\sigma_{\lambda_1/2 }}{\sqrt{3}}[/tex]

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.

[tex]\lambda = 2 \cdot \lambda_1/2 \pm 2 \cdot \frac{\sigma_{\lambda_1/2 }}{\sqrt{3}} [/tex]

Third.

[tex]v = \lambda \cdot f [/tex]

[tex] \overline{v} = \overline{\lambda} \cdot \overline{f} [/tex]

[tex]\left | \frac{\Delta v }{v} \right |= \sqrt{\left | \frac{\Delta \lambda }{\lambda} \right |^2+\left | \frac{\Delta f }{f} \right |^2} [/tex]

So now, I have three values for the speed and am looking for their average value. Are the data sets consistent?
 
  • #4
I've just had a very simple idea for a solution.

I have three values

v1=v1_mean+- v1_error
v2=v2_mean+- v2_error
v3=v3_mean+- v3_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
 
  • #5
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

[tex]\bar x = \frac{\sum_i x_i/\sigma_i^2}{\sum_i 1/\sigma_i^2}[/tex]

with those σi 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.
 

1. How do I find the mean value from multiple absolute error data?

To find the mean value from multiple absolute error data, you can add all of the absolute error values together and then divide by the number of data points. This will give you the average absolute error.

2. What is the purpose of finding the mean value from multiple absolute error data?

The mean value from multiple absolute error data gives an overall representation of the accuracy of the data points. It can also help in identifying any outliers or inconsistencies in the data.

3. Is it necessary to calculate the mean value from multiple absolute error data?

It is not always necessary, but it can provide useful information about the accuracy of the data. If you are looking for a general understanding of the data, calculating the mean absolute error can be helpful.

4. Can the mean value from multiple absolute error data be negative?

No, the mean value from multiple absolute error data cannot be negative. Absolute error is always a positive value, so when calculating the mean, it will also result in a positive value.

5. How do I interpret the mean value from multiple absolute error data?

The mean value from multiple absolute error data gives an indication of the average distance between the measured values and the true values. A lower mean value indicates a more accurate data set, while a higher mean value indicates more variability and potential errors in the data.

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