Why is chi^2/ndf close to 1 a good fit?

  • Thread starter DougUTPhy
  • Start date
  • Tags
    Fit
This is a measure of how well the model fits the data.In summary, a \chi^2/\mathrm{ndf} (number of degrees of freedom) close to one indicates a good fit in a statistical model, as it represents a sum of squares of standardized normal random variables, with each fraction being close to one. This measure is used to determine how well the model fits the data.
  • #1
DougUTPhy
11
0
Why is [itex]\chi^2 / \mathrm{ndf}[/itex] (number of degrees of freedom) close to one mean that a fit is a good fit?I have had this question for a long time, and now I'm currently in a lab where the instructor and TA's love to see you talk about [itex]\chi ^2[/itex] -- so it's killing me! All I have ever heard is that it is a good fit, but I have never heard why. Or what the difference is between a being a little above or a little below one.

I hope math is a good board to put this in, I kind of feel like it's a statistics question.
Just a general question to quench my curiosity...
Thanks for any insight!
 
Mathematics news on Phys.org
  • #2
The statistic has the form
[tex]
\chi^2 = \sum_{i=1}^{n} \frac{(X_i - \mu_i)^2}{\sigma_i^2},
[/tex]
i.e., it is a sum of squares of standardized normal random variables. If your fit is good, i.e. [itex]\mu_i[/itex] and [itex]\sigma_i^2[/itex] are well estimated, you suppose each fraction to be close to one. Hence the sum gives [itex]n[/itex] and therefore [itex]\chi^2/n[/itex] gives a number close to 1.
 

FAQ: Why is chi^2/ndf close to 1 a good fit?

Why is it important for chi^2/ndf to be close to 1 in a good fit?

The chi^2/ndf (chi-squared per degree of freedom) is a measure of the goodness of fit in statistical models. It represents the ratio of the observed data to the expected data, and a value close to 1 indicates that the model fits the data well. This is important because it means that the model is accurately representing the relationship between the variables and is not overfitting or underfitting the data.

2. How is chi^2/ndf calculated and what does it represent?

The chi^2/ndf is calculated by taking the sum of the squared differences between the observed and expected data, divided by the number of degrees of freedom in the model. It represents the amount of discrepancy between the observed data and the expected data, with a lower value indicating a better fit.

3. Can chi^2/ndf be used to compare different models?

Yes, chi^2/ndf can be used to compare different models. A lower value of chi^2/ndf indicates a better fit, so the model with the lowest chi^2/ndf can be considered the best fit for the data.

4. Are there any limitations to using chi^2/ndf as a measure of goodness of fit?

Yes, there are some limitations to using chi^2/ndf as a measure of goodness of fit. It assumes that the data follows a normal distribution, and it can be affected by outliers in the data. Additionally, it does not take into account the complexity of the model, so a low chi^2/ndf value may not necessarily indicate the best model.

5. How can chi^2/ndf be improved in model fitting?

To improve chi^2/ndf in model fitting, it is important to carefully select the appropriate model for the data and to address any outliers or non-normality in the data. Additionally, using alternative goodness of fit measures such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC) can provide a more comprehensive evaluation of the model's performance.

Similar threads

Best goodness of fit test for low expected values
Replies
11
Views
2K
A The chi square goodness-of-fit test with no degrees of freedom left
Replies
6
Views
1K
What is the correct formula for the reduced Chi square?
Replies
3
Views
2K
Which Fit Should Be Chosen When Goodness of Fit Values Are Close?
Replies
3
Views
2K
How to set up and interpret Chi^2 test results for my data?
Replies
11
Views
1K
Can an n-dimensional object fit entirely in n-1 dimensions?
Replies
2
Views
2K
Goodness of fit, Residual STD, chi square
Replies
1
Views
3K
I About chi-squared and r-squared test for fitting data
Replies
5
Views
8K
I How do I determine if improving a fit is significant?
Replies
13
Views
1K
I Level of significance and acceptance and rejection of the null hypothesis
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top