Relating chi-squared and gaussian curves

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SUMMARY

The chi-squared distribution can be represented as a Gaussian distribution under certain conditions, specifically when the chi-squared distribution is suitably standardized. This relationship is grounded in the fact that the chi-squared distribution is derived from the sum of squares of independent Gaussian random variables (RVs) with mean 0 and unit variance. As the degrees of freedom increase, the approximation to a normal distribution improves. However, it is crucial to differentiate between chi-squared distributions and chi-squared tests, as the latter depend heavily on data categorization and bin sizes.

PREREQUISITES
  • Understanding of chi-squared distribution and its properties
  • Knowledge of Gaussian random variables (RVs)
  • Familiarity with statistical tests, particularly Goodness of Fit tests
  • Concept of degrees of freedom in statistical distributions
NEXT STEPS
  • Study the relationship between chi-squared distributions and Gaussian distributions in detail
  • Learn about the implications of degrees of freedom on statistical approximations
  • Explore the application of Goodness of Fit tests in various statistical analyses
  • Investigate the characteristics of non-central chi-squared distributions
USEFUL FOR

Statisticians, data analysts, and researchers involved in statistical modeling and hypothesis testing will benefit from this discussion, particularly those working with chi-squared and Gaussian distributions.

Ai52487963
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Simple question: can a chi-squared be represented as a gaussian distribution? I'm wondering if I can take some chi-squared numbers that I have and represent them as increasing/decreasing widths of FWHM of a gaussian. Can I?
 
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Ai52487963 said:
Simple question: can a chi-squared be represented as a gaussian distribution? I'm wondering if I can take some chi-squared numbers that I have and represent them as increasing/decreasing widths of FWHM of a gaussian. Can I?

The Chi Sq distribution is the sum of the squares of a set of independent Gaussian RVs with mean 0 and unit variance. However, a generalized Chi Sq distribution can be derived from Gaussian RVs with non-zero means and non unit variance. This is the basis of Goodness of Fit tests. So the answer to your question is generally yes. Note you must distinguish between the Chi Sq distribution and Chi Sq tests which are highly dependent on how the data is categorized and on relative bin sizes.
 
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Suitably standardized, probabilities from a central chi-square distribution can be approximated by a normal distribution - the approximation improves as the number of degrees of freedom increases.

A similar result can hold for the non-central (I assume this is what SW meant be generalized) chi-square distribution.

However, each individual case must be investigated on its own; the problem is (obviously) that central chi-square distributions are right-skewed, and have a natural boundary at zero, while the normal distribution has neither of these properties.
 

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