Statistical Error of Centroid of Gaussian Distribution

In summary, the "origin estimation error" in a Gaussian distribution can be expressed as a function of the number of data samples, and as the number of samples increases, the error decreases according to a 1/sqrt(L) relationship. A proof for this behavior can be found in J.R. Taylor's book on error analysis.
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Statistical "Error" of Centroid of Gaussian Distribution

If I have L data samples, distributed randomly (3D real Gaussian distibution, unity variance) about the origin in 3D real space, how can I derive an expression for the "origin estimation error" (i.e. the distance between the true origin and the centroid of the data points) as a function of L?

Intuitively, as L->infinity, the error->0. In fact, it is easy to show in Matlab that the error falls as 1/sqrt(L) (for sufficiently large number of trials). However, I don't know where to start with a proof. (I'm really trying to write a proof for N-dim complex space, but I expect that will only need an extra sqrt(2) term).

Any advice is much appreciated!
 
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Problem solved:

Of course, I'm simply looking for the standard deviation of the mean. A proof for its behaviour as a function of number of samples can be found in:

J.R. Taylor, "An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements", pp. 147-148.
 

What is the statistical error of centroid of Gaussian distribution?

The statistical error of centroid of Gaussian distribution refers to the uncertainty or variation in the calculated mean or central point of a Gaussian distribution. It is a measure of how much the calculated centroid may deviate from the true centroid of the distribution.

How is the statistical error of centroid of Gaussian distribution calculated?

The statistical error of centroid of Gaussian distribution is typically calculated using the formula:
σ = √(∑(xi - μ)^2 / n)
where σ is the standard deviation, μ is the mean, xi is each data point, and n is the total number of data points.

What factors can affect the statistical error of centroid of Gaussian distribution?

The statistical error of centroid of Gaussian distribution can be affected by a variety of factors, such as the sample size, the shape and spread of the distribution, and the accuracy of the data points. It can also be influenced by any outliers or extreme values in the data.

Why is it important to consider the statistical error of centroid of Gaussian distribution?

Considering the statistical error of centroid of Gaussian distribution is important because it allows for a better understanding of the accuracy and reliability of the calculated centroid. It also helps in making decisions and drawing conclusions based on the data.

How can the statistical error of centroid of Gaussian distribution be reduced?

The statistical error of centroid of Gaussian distribution can be reduced by increasing the sample size, ensuring the data is accurate and representative of the overall population, and identifying and addressing any outliers or extreme values that may impact the centroid calculation. Using more precise measurement techniques can also help in reducing the error.

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