Chi-squared and reduced Chi-squared

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In summary, the Chi-squared and reduced Chi-squared equations are used in a physics lab to determine the "goodness of fit" of data. The book states that the chi-squared value should be close to zero and the reduced chi-squared value should be close to 1, but the meaning of these values is unclear since chi-squared is divided by the number of degrees of freedom. The reduced chi-squared distribution is not fully understood, but it is intuitive that the variation between expected and observed should be minimized. More information is needed to fully understand the significance of these values.
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Liquidxlax
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In my physics honours lab to check our "goodness of fit" we have to use the Chi-squared and reduced Chi-squared equations.

For the Lab we use the text Bevington Data analysis and error reduction. I'm having a problem on determining whether or not my chi-squared values are good because the book states that chi-squared should be close to zero while the reduced chi-squared should be close to 1.

this obviously doesn't make sense because you divide chi-squared by the number of degrees of freedom which is a positive integer number.

Can somebody please explain to me the meaning of both reduced and chi-squared numbers?
 
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Liquidxlax said:
In my physics honours lab to check our "goodness of fit" we have to use the Chi-squared and reduced Chi-squared equations.

For the Lab we use the text Bevington Data analysis and error reduction. I'm having a problem on determining whether or not my chi-squared values are good because the book states that chi-squared should be close to zero while the reduced chi-squared should be close to 1.

this obviously doesn't make sense because you divide chi-squared by the number of degrees of freedom which is a positive integer number.

Can somebody please explain to me the meaning of both reduced and chi-squared numbers?

I am not sure about the reduced chi-squared, but since you are measuring variation between expected and observed, it makes sense intuitively that the variation should be minimized (and hence close to zero).

Can you give the definition of a reduced chi-square distribution and any sample statistics that you have to calculate?
 

1. What is Chi-squared and how is it used in statistics?

Chi-squared is a statistical test used to determine if there is a significant relationship between two categorical variables. It compares the observed frequencies of these variables to the expected frequencies under the assumption of independence. This allows us to determine if the variables are related or if any observed differences are due to chance.

2. How is Chi-squared calculated?

The Chi-squared statistic is calculated by taking the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies. This results in a single number that indicates the degree of relationship between the variables.

3. What is reduced Chi-squared and how is it different from regular Chi-squared?

Reduced Chi-squared is a modified version of the Chi-squared test that takes into account the number of degrees of freedom in the data. This is important because as the number of variables or categories increases, the expected frequencies also increase, making the Chi-squared statistic larger. Reduced Chi-squared normalizes the Chi-squared statistic by dividing it by the number of degrees of freedom, making it easier to compare results across different data sets.

4. When should I use Chi-squared vs. reduced Chi-squared?

Chi-squared is typically used when analyzing data with two categorical variables and is used to determine if there is a significant relationship between them. Reduced Chi-squared is used in situations where there are more than two categorical variables or when comparing results from different data sets.

5. What are the limitations of using Chi-squared and reduced Chi-squared?

One limitation is that Chi-squared can only be used for categorical data and not continuous data. Additionally, Chi-squared assumes that the expected frequencies are greater than 5 for all categories, so it may not be appropriate for small sample sizes. Reduced Chi-squared also has limitations in terms of interpreting the results, as it can only indicate if there is a relationship between variables, but not the direction or strength of the relationship.

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