Error Propogation: Calculating Average Acceleration w/ Uncertainty

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In summary, this isn't a homework problem. I'm just having ALOT of trouble understanding how to do error propogation.
  • #1
012anonymousx
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This isn't a homework problem. I'm just having ALOT of trouble understanding how to do error propogation.
[EDIT] These are actually lab results. But its not a "homework" question. Its how to do uncertainty. Just trying to be clear.

Consider we take different measurements of an objects acceleration with uncertainties:
-0.2590 ± 0.0065 m/s^2
-0.2760 ± 0.0019 m/s^2
-0.2800 ± 0.0057 m/s^2
-0.2510 ± 0.0230 m/s^2
-0.2640 ± 0.0073 m/s^2
I want to take the average acceleration with the uncertainty of the average.
So the average is fine and easy however;
There are two ways to take the uncertainty:
1. Standard deviation of the acceleration divided by the square root of the number of accelerations.
2. Since to get the mean accelerate we summed accelerations, we use standard error propogation rules for addition and sum the square of the uncertainties and take their square root.

Notice in both of the above, we ignore the other method. In the first, we ignore the fact that each value has its own uncertainty and just look at the standard deviation.

In the second, we ignore the distribution of our data and only look at their individual uncertainties.

So what do I do...? :(.
 
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  • #2
012anonymousx said:
So the average is fine and easy however.
Is it? It's not. One thing you don't want to do is to let that noisy measurement of -0.2510 have near as much influence as those other measurements, and that very clean measurement of -0.2760 should have a lot more influence than any of the other measurements. There are lots of ways to do this, but one widely used approach is to weight the measurements by the inverse of the variance. It's a lot easier if the weights sum to one:
$$w_i = \frac{1/\sigma_i^2}{\sum_j 1/\sigma_j^2}$$
With this, the weighted mean is
$$\bar x = \sum w_i x_i$$

What about the variance? An estimate of the variance that follows the same paradigm is given by
$$s^2 = \sum w_i^2 \sigma_i^2$$
Note: This is a biased estimate. It underestimates the variance, with the bias most marked with a small sample size.
 
  • #3
Ugh... I'm just a first year. My statistics class is next year and I only have to follow standard rules for now, so even if I wanted to, I probably shouldn't look up formulas, especially when there are apparently more than one way to do something (as per yourself).
Also, why should one thing have extra weight and the other less? That seems like we are biasing the results and doesn't make a lot of sense to me. So I will refrain.

Back to my original problem. How do I go about calculating the final uncertainty of the unweighted average?

I really really appreciate your response though.
 

1. What is error propagation?

Error propagation is the process of estimating the uncertainty or error in a calculated value based on the uncertainties or errors in the measured quantities used in the calculation. It is a way to quantify the level of confidence we have in our calculated results.

2. How is error propagation used in calculating average acceleration?

In calculating average acceleration, we use error propagation to determine the uncertainty in the measured values of initial velocity, final velocity, and time. This uncertainty is then used to calculate the uncertainty in the average acceleration using a formula that takes into account the uncertainties in all three measured quantities.

3. Why is it important to consider error propagation in scientific calculations?

It is important to consider error propagation in scientific calculations because it allows us to account for the inherent uncertainty in our measurements and ensures that our calculated results reflect the true level of confidence we have in them. Ignoring error propagation can lead to incorrect conclusions and inaccurate data.

4. What factors can affect the uncertainty in a calculated value?

The uncertainty in a calculated value can be affected by the uncertainties in the measured quantities used in the calculation, as well as any assumptions or approximations made in the calculation. It can also be influenced by external factors such as equipment limitations or human error.

5. How can error propagation be minimized?

Error propagation can be minimized by using precise and accurate measurement techniques, reducing the number of assumptions and approximations in the calculation, and by repeating the experiment multiple times to obtain a more precise average value. Additionally, using appropriate units and significant figures can also help to minimize error propagation.

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