Relating chi-squared and gaussian curves

In summary, a chi-squared can potentially be represented as a Gaussian distribution, but it depends on the specific case and data being used. Goodness of Fit tests are often based on this concept, but it is important to consider the differences in properties between the two distributions.
  • #1
Ai52487963
115
0
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?
 
Physics news on Phys.org
  • #2
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.
 
Last edited:
  • #3
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.
 

FAQ: Relating chi-squared and gaussian curves

1. What is the chi-squared test and how is it related to gaussian curves?

The chi-squared test is a statistical test used to determine whether there is a significant difference between the expected and observed frequencies of data. It is often used to test the goodness of fit of a model to observed data. Gaussian curves, also known as normal curves, are commonly used to model continuous data and can be used in the chi-squared test to determine the expected frequencies.

2. How do you calculate the chi-squared value for a gaussian curve?

The chi-squared value for a gaussian curve is calculated by summing the squared differences between the observed and expected frequencies, divided by the expected frequency for each category. This calculation is then compared to a critical value from a chi-squared distribution to determine the significance of the results.

3. Can the chi-squared test be used with any type of data?

No, the chi-squared test is most commonly used with categorical or discrete data. However, it can also be used with continuous data if the data is grouped into categories.

4. How can the results of a chi-squared test be interpreted for a gaussian curve?

If the chi-squared value calculated is greater than the critical value, it indicates that there is a significant difference between the expected and observed frequencies, and the gaussian curve does not fit the data well. If the chi-squared value is smaller than the critical value, it suggests that the gaussian curve fits the data well.

5. Is it possible to have a perfect fit between a gaussian curve and observed data?

In theory, it is possible to have a perfect fit between a gaussian curve and observed data. However, in practice, there is always some level of error or deviation from the expected frequencies, and a perfect fit is rare. The aim of the chi-squared test is to determine whether the level of deviation is significant enough to reject the fit of the gaussian curve to the data.

Back
Top