University of Utah
Question
Asked 27 March 2014
What number of points is required to fit Gaussian Distributions?
A Gaussian distribution has an exponential term and 2 variables. Its log will turn it linear. But I am not able to figure out how many points will suffice to fit a gaussian distribution and why? It is recommended to be 5 to 12 points in some articles but no concrete reasons are given.
Most recent answer
@Deepak Kumar Akar,
Sorry for the delay in responding. As discussed earlier, if you already know that the curve is Gaussian then the problem is much easier. In any case, it depends on how accurately you would like to characterize the distribution. For example, if you are interested in the mean of the distribution then, statistically speaking, the certainty of the determination improves proportionally to the square root of the number of points. For example, if you acquire 16 points then the uncertainty in the mean is half as much as if you acquire 4 points. I don't recall what the relationship is for the standard deviation, i.e. how it improves as the number of sampling points increases.
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Popular answers (1)
University of Utah
If there is no error in the points then thee unique points will determine the fit. This is because there are three degrees of freedom that in the function being fitted, so three points are needed to fit the data.
If there is no error in the points and you somehow knew beforehand that the underlying distribution were normalized to unit probability then two points are enough to uniquely determine the distribution.
In making the comments above seveal assumptions are made. One is that there is no error in the data. This assumption was already stated.
The most important underlying assumption is that the curve you are trying to fit actually is Gaussian.
Also, an assumption is made regarding the meaning of "points". In the discussion above the term "points" is taken to mean that the value of the underlying curve is known at two or three points. For example, if your friend calculated three points on a Gaussian curve and handed the results to you, that would be an example where you would know the value of three points on the curve.
Another case would be if you or someone else were generating a set of random data taken from a Gaussian distribution. This could be via a computer-generated Monte-Carlo sampling from a Gaussian distribution, or from a set of experimentally generated data, or some other statistically generated method. The "points" in this case have an entirely different meaning. They are now repeated draws from a Gaussian distribution. In this case the number of points required is much larger, and it depends on how close you need to be to the "true" values. For example, the statistical error in the mean of the distribution varies as one over the square root of the number of sample points.
I have given an incomplete discussion, but it might be enough to get you started on the right track.
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All Answers (8)
University of Utah
If there is no error in the points then thee unique points will determine the fit. This is because there are three degrees of freedom that in the function being fitted, so three points are needed to fit the data.
If there is no error in the points and you somehow knew beforehand that the underlying distribution were normalized to unit probability then two points are enough to uniquely determine the distribution.
In making the comments above seveal assumptions are made. One is that there is no error in the data. This assumption was already stated.
The most important underlying assumption is that the curve you are trying to fit actually is Gaussian.
Also, an assumption is made regarding the meaning of "points". In the discussion above the term "points" is taken to mean that the value of the underlying curve is known at two or three points. For example, if your friend calculated three points on a Gaussian curve and handed the results to you, that would be an example where you would know the value of three points on the curve.
Another case would be if you or someone else were generating a set of random data taken from a Gaussian distribution. This could be via a computer-generated Monte-Carlo sampling from a Gaussian distribution, or from a set of experimentally generated data, or some other statistically generated method. The "points" in this case have an entirely different meaning. They are now repeated draws from a Gaussian distribution. In this case the number of points required is much larger, and it depends on how close you need to be to the "true" values. For example, the statistical error in the mean of the distribution varies as one over the square root of the number of sample points.
I have given an incomplete discussion, but it might be enough to get you started on the right track.
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Gauhati University
Actually, there cannot be a definite answer to this question! Not just the Gaussian probability law, for every other probability law, it cannot be said that to fit the law, we need certain prefixed number of observations,. Indeed, the more, the better.
If the number of observations is small, to fit the Gaussian law, or any other law for that matter, it is better to use the method of successive approximation using the Taylorian expansion.
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SCHWIND eye-tech-solutions GmbH
In some other papers you can find as a general rule a minimum of 20 observations per each degree of freedom (sometimes DoF + 1), this will make 60 (or 80) observations for a robust fit. Why 20, is not clear to me, I will check my secondary references.
University of Utah
Again, I think it is important to clarify what is meant by "point". If one is sampling off of a mathematically-defined curve (e.g. calculating the value of a function at several points) then two points are enough to completely define a normalized Gaussian distribution, and three points are enough to completely define an unnormalized Gaussian distribution.
If one is taking a series of experimental points (or doing a Monte Carlo sampling of a probability distribution on a computer) and attempting to determine the statistics of the data set generated experimentally, then one is dealing with a different problem, and the answer will depend on the accuracy required.
To determine whether a distrubiton is Gaussian there are a number of different statistical tests that can be applied. One is known as the Anderson-Darling test. There are others as well.
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NIN, Hyderabad
It could be 5 observations for each continuous class interval. It may be enough to have at least 30 observations to fit a normal distribution. Just try a simulation study, you will come to know the difference.
Bhabha Atomic Research Centre
@Alan Rockwood I want to add that the points have been generated by some experiment (these points may have some inherent fluctuation and the fluctuation from point to point but is fixed for any particular point)and it is known that the points will fall on a guassian curve. For example photopeak in gamma ray spectrometry.
University of Utah
@Deepak Kumar Akar,
Sorry for the delay in responding. As discussed earlier, if you already know that the curve is Gaussian then the problem is much easier. In any case, it depends on how accurately you would like to characterize the distribution. For example, if you are interested in the mean of the distribution then, statistically speaking, the certainty of the determination improves proportionally to the square root of the number of points. For example, if you acquire 16 points then the uncertainty in the mean is half as much as if you acquire 4 points. I don't recall what the relationship is for the standard deviation, i.e. how it improves as the number of sampling points increases.
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