# What is the correct formula for the reduced Chi square?

• patric44
In summary, the conversation discusses the calculation of reduced Chi square and root mean square deviation (RMSD) for a set of data points. There is confusion regarding which formula is the correct one, as some sources refer to it as RMSD while others call it reduced chi square. The conversation also delves into the concept of minimizing reduced chi square for the best fit, with some confusion about the optimum value being 1 or 1-reduced chi square. The elements of the formulae are carefully defined, with ##y## representing the measured data, ##\tilde{y}## being the calculated data from a specific model, ##\delta y_i## being the error in measuring ##y##, and ##m## being
patric44
Homework Statement
what is the correct formula of reduced Chi square
Relevant Equations
\Chi^2
Hi all
I want to calculate the reduced Chi square and root mean square deviation RMSD of some data points that i have, but I am confused about the correct formula for each of them, which one is the correct one. I found this formula in a paper where they referred to it as the RMSD :
$$\chi=\sqrt{\frac{1}{N}\sum_{i}^{N}\left(\frac{(y_{i}-\tilde{y}_{i})}{\delta y_{i}}\right)^{2}}$$
and in some books the same formula with little modification (instead of ##N## they put the degrees of freedom) as :
$$\chi=\sqrt{\frac{1}{N-m}\sum_{i}^{N}\left(\frac{(y_{i}-\tilde{y}_{i})}{\delta y_{i}}\right)^{2}}$$
which one is reduced ##\chi^{2}## and which is RMSD if any of them?!
another question why i read that we need to minimize the value of reduced ##\chi^{2}## to get the best fit, isn't the optimum value is 1 ?! , shouldn't we minimize 1-##\chi^{2}## or what?
I will appreciate any help, thanks in advance

Please carefully define the elements in these formulae, particularly ##\tilde{y}_{i}## and
##\delta y_i ## and what is m?)

hutchphd said:
Please carefully define the elements in these formulae, particularly ##\tilde{y}_{i}## and
##\delta y_i ## and what is m?)
##y## is the measured data
##\tilde{y}## is the calculated data from a specific model
##\delta y_i ## is the error in measuring ##y##
##m## the number of parameters of the model
I am not talking about the so called category chi2. I mean the other one

hutchphd
I think the formula with the m is appropriate. Very often m=1 when the the mean value is taken as a "fitted" parameter from the data. I have no idea about the names and categories of these things sorry.

patric44

## 1. What is the reduced Chi square formula used for?

The reduced Chi square formula is used to evaluate the goodness of fit of a statistical model to a set of observed data. It is commonly used in hypothesis testing and model comparison.

## 2. What is the correct formula for the reduced Chi square?

The correct formula for the reduced Chi square is χ² = Σ[(O-E)²/E], where χ² is the reduced Chi square, O is the observed value, and E is the expected value.

## 3. How is the reduced Chi square different from the regular Chi square?

The reduced Chi square is a modified version of the regular Chi square that takes into account the degrees of freedom in the model. It is calculated by dividing the regular Chi square by the degrees of freedom, resulting in a smaller value that is easier to interpret.

## 4. When should the reduced Chi square formula be used?

The reduced Chi square formula should be used when evaluating the goodness of fit of a model with a large number of parameters. It is also commonly used when comparing multiple models to determine which one best fits the observed data.

## 5. What does a reduced Chi square value of 1 indicate?

A reduced Chi square value of 1 indicates that the model fits the observed data well and there is no significant difference between the expected and observed values. Generally, a value close to 1 (between 0.8 and 1.2) is considered a good fit, while values greater than 1 indicate a poor fit.

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