Statistical uncertainty of weighted mean

In summary, Brais is using physical data to do an analysis that involves ~30k measurements of energies, momenta, and angles of particles. They calculate a value, v, at the end and get v±σ if all the data is inputted into their program. However, if they "bin" their events in energy and angle and calculate v, they get v1±σ1, v2±σ2, v3±σ3, v4±σ4. They then combine these values using a weighted average and get v_c and σ_c, where σ_c is approximately equal to σ/2. Brais is seeking an explanation for this result and mentions using a formula from a Wikipedia article, but is unsure
  • #1
Brais
7
0
Hello!

I am using physical data to do an analysis (~30k measurements). These measurements include energies, momenta, angles... of particles.
I am calculating a value (call it v) at the end after a lengthy process, and if I introduce all the data into my program I did, the result is v±σ.
If, however, I "bin" my events in energy and angle (say I made four bins in total), when I calculate "v", I get v1±σ1, v2±σ2, v3±σ3, v4±σ4. Then I combine these values into one using a weighted average (c stands for combined): [itex] v_c = \sum(v_i/\sigma^2_i)/\sum(1/\sigma^2_i)[/itex], and [itex]\sigma_c = 1/\sqrt{\sum(1/\sigma^2_i)}[/itex] (as can be seen here).
When I do this, it turns out that [itex]\sigma_c \simeq \sigma/2[/itex]. How can this be? I am using the same amount of statistics!

Any reply or idea will be very welcome!

Thank you!

Brais.
 
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  • #2
Hey Brais.

The wiki article looks straight-forward, but perhaps you could just outline your calculations in a little more detail step by step to show the simplifications and assumptions you used.
 
  • #3
Hi, thanks for your reply!
I am calculating a fit. If I put all my data together I get an error that is higher than that of fitting different sets of points separately and then combining them with a weighted mean.
I didn't do any simplification, just applied the expression seen in wikipedia.

Brais
 
  • #4
Brais said:
I didn't do any simplification, just applied the expression seen in wikipedia.
Brais

You are using a formula from that article that applies when you know the standard deviations of the distributions that are involved, but I'd guess that you don't. Your [itex] \sigma_i [/itex] are probably estimators of standard deviations that you computed from the sample. (The term "standard deviation" is ambiguous. It has at least 5 different meanings in statistics, depending on the context where it appears.)
 
  • #5
Following Stephen Tashi's post, you should probably just clarify exactly what attribute you are using.
 
  • #6
Hi again!

A long time ago I had to stop this analysis and so my doubt wasn't importantr for some time :)
I use the errors that my minimization algorithm "MINUIT" gives. Unfortunatelly I cannot find anything except that it (obviously) calculates a covariance matrix and error matrix...

Thanks,

Brais
 

1. What is statistical uncertainty of weighted mean?

Statistical uncertainty of weighted mean is a measurement that describes the level of uncertainty or variability in the calculated weighted average of a set of data points. It takes into account both the individual data points and their associated weights to provide a more accurate representation of the overall average.

2. How is statistical uncertainty of weighted mean calculated?

The statistical uncertainty of weighted mean is calculated using the standard deviation formula, taking into account the weights of each data point. This formula is more complex than the standard deviation formula for unweighted data, as it involves the calculation of the weighted mean and the sum of squared deviations from the weighted mean.

3. Why is statistical uncertainty of weighted mean important?

The statistical uncertainty of weighted mean is important because it provides a measure of the precision and reliability of the calculated weighted average. It is especially useful when comparing different sets of data with varying sample sizes and weights, as it allows for a more accurate comparison of the central tendency of the data.

4. How does increasing the sample size affect statistical uncertainty of weighted mean?

Increasing the sample size typically decreases the statistical uncertainty of weighted mean. This is because a larger sample size provides more data points and reduces the impact of outliers, resulting in a more precise and accurate calculation of the weighted mean.

5. Can statistical uncertainty of weighted mean be negative?

No, statistical uncertainty of weighted mean cannot be negative. This is because the standard deviation, which is used to calculate the uncertainty, is always non-negative. A negative value would not make sense in the context of measuring the variability of data points around a central average.

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