# Deg. of Freedom Q and Red. Chi^2

• FortranMan
In summary, the speaker is struggling to determine the degrees of freedom for a data set that has around 40 observations. The data set consists of 4 sets of 10 observations each, measuring at different temperatures and magnetic fields. The speaker believes the degrees of freedom to be 1 since the observations are distinct. They are also aware of the reduced chi^2 value and its significance when comparing experimental data to theoretical models in physics. They are unsure of what value would be considered "good" and if there is a specific academic source to support this value. The speaker is conducting a test to compare field variation in superconductors versus temperature and field, and is comparing a graph of the data to a theoretical graph generated by the function sigma(T
FortranMan
I'm having trouble determining with confidence the deg.s of freedom (df) of a data set I'm dealing with.

I'm dealing with a data set of around 40 observations. 4 sets of 10 observations measure at different temperatures from zero to T (of some interval delta), each set representing a different magnetic field B, giving as an output a value sigma, such that theoretically it should be described by a function sigma(T,B) (the null hypothesis). Since these observations are all distinct, I am guessing the df = 1, right?

Also, I am aware the reduced chi^2 (r-chi^2) value is (chi^2)/df, and that a r-chi^2 value >> 1 or << 1 is bad. What r-chi^2 value is considered a "good" value when comparing experimental data to theoretical models (specifically in phyiscs)? Is there a specific academic source that confirms or supports this value?

I'm having trouble determining with confidence the deg.s of freedom (df) of a data set I'm dealing with.
What is the test you are doing? What are the inputs to the test statistic?

A hypothesis is a mathematical statement, so "sigma(T,B)" cannot be the null hypothesis. Did you mean sigma(T,B) = 0?

EnumaElish said:
What is the test you are doing? What are the inputs to the test statistic?

Field variation in superconductors verses temperature and field. I am comparing a graph of that data to a theoretical graph which would hopefully describe the data (satisfying the null hypothesis in that there is no great difference between the data and the model).

A hypothesis is a mathematical statement, so "sigma(T,B)" cannot be the null hypothesis. Did you mean sigma(T,B) = 0?

sigma(T,B) is the function that generates the theoretical graph.

## 1. What is the concept of degrees of freedom in statistical analysis?

Degrees of freedom refer to the number of independent pieces of information available in a data set. It is the number of values that are free to vary when calculating a statistic. In other words, it represents the number of observations that are independent and not restricted by any constraints. In statistical analysis, degrees of freedom play a crucial role in determining the accuracy and reliability of the results.

## 2. How is degrees of freedom related to the chi-square test?

In the chi-square test, degrees of freedom are determined by subtracting the number of constraints from the total number of categories being compared. For example, if we are comparing the results of a survey with 5 possible options, the degrees of freedom will be 5-1=4. This value is then used to determine the critical value for the chi-square statistic, which is used to determine the statistical significance of the results.

## 3. What is the reduced chi-square statistic and how is it calculated?

The reduced chi-square statistic is a measure of how well a statistical model fits the data. It is calculated by dividing the chi-square statistic by the degrees of freedom. This value is then compared to a critical value to determine if the model is a good fit for the data. A lower reduced chi-square value indicates a better fit, while a higher value suggests that the model does not fit the data well.

## 4. What are the limitations of using reduced chi-square as a measure of model fit?

The reduced chi-square statistic can be affected by the sample size, with larger samples resulting in lower values. This can make it difficult to compare models with different sample sizes. Additionally, it assumes that the data follows a specific distribution, which may not always be the case in real-world scenarios.

## 5. How can I interpret the results of a chi-square test with degrees of freedom?

If the chi-square test results in a p-value that is less than the predetermined significance level (usually 0.05), then we can reject the null hypothesis and conclude that there is a significant relationship between the variables being compared. The degrees of freedom can also help us determine the strength of this relationship, with a higher number of degrees of freedom indicating a stronger relationship between the variables.

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