Finding reduced chi^2 for independent variable?

Your name]In summary, the individual is having difficulty calculating the reduced chi-squared value for a function fit to experimental data due to large errors in the independent variable. They have attempted to switch the independent and dependent variables, but are looking for a better solution. Two potential solutions are suggested: using a weighted least squares fitting method or a Monte Carlo simulation, both of which take into account the errors in both variables.
  • #1
khkwang
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Homework Statement


I have a function V = kI
where k is some constant
I_err = 0.005 A
V_err = 0.00005 V

A fit was then made, but a problem occurs when I try to calculate the reduced chi^2.
Since the error of the dependent variable V is so small, the resultant reduced chi^2 is fairly large; with good reason because the vertical error bars clearly don't overlap with the fit data.

However, the large error in the independent variable I is large enough to compensate and so the data agrees with the fit within error. If I flip the independent and dependent variables, I get a perfect reduced chi^2. However for the purposes of the fit, it would be strange to do so.

Is there a way to include the independent variable in the calculation for reduced chi^2?


Homework Equations



The equation I'm using right now is:
chi^2 = sum(((V_exp - V_fit) / V_err)^2)
reduced chi^2 = chi^2/((# of values in V_exp) - (# of parameters))


The Attempt at a Solution



No attempt thus far except switching independent and dependent variables. Which gives a good reduced chi-squared but is not what I want.
 
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  • #2


Hello there,

Thank you for sharing your problem with us. It seems like you are trying to fit a function to experimental data and are having trouble calculating the reduced chi-squared value due to large errors in the independent variable. In this case, it may be helpful to use a weighted least squares fitting method, which takes into account the errors in both the dependent and independent variables.

The weighted least squares method involves assigning weights to each data point based on the magnitude of its error. In your case, since the error in the independent variable is larger, those data points should have a higher weight compared to the data points with smaller errors. This will help to balance out the effect of the large error in the independent variable on the chi-squared value.

Another option could be to use a different fitting method, such as a Monte Carlo simulation, which also takes into account the errors in both variables. This method involves generating a large number of data sets with random variations based on the errors in the original data, and then fitting the function to each data set. The average of the fitted parameters from all the simulations can then be used as the final result.

I hope this helps. Let me know if you have any further questions or if you need any clarification.


 

1. What is reduced chi^2 and why is it important in scientific research?

Reduced chi^2 is a statistical measure used to evaluate the goodness of fit of a model to a set of data. It is important in scientific research because it allows researchers to determine how well a model fits their data, and whether any discrepancies between the model and the data are significant or due to random chance.

2. How is reduced chi^2 calculated?

Reduced chi^2 is calculated by dividing the chi^2 value (sum of squared differences between the observed data and the model predictions) by the number of degrees of freedom. The degrees of freedom are equal to the number of data points minus the number of parameters in the model.

3. What does a reduced chi^2 value of 1 indicate?

A reduced chi^2 value of 1 indicates that the model fits the data well and there are no significant discrepancies between the two. This is considered a good fit and is the goal when using this statistical measure.

4. What does a reduced chi^2 value greater than 1 indicate?

A reduced chi^2 value greater than 1 indicates that the model does not fit the data well and there are significant discrepancies between the two. This could mean that the model is not a good representation of the data, or that there are other factors that are influencing the results.

5. How can I use reduced chi^2 to compare different models?

Reduced chi^2 can be used to compare different models by calculating the value for each model and choosing the one with the lowest value. A lower reduced chi^2 value indicates a better fit to the data. However, it is important to also consider the number of parameters in each model, as a model with more parameters may have a lower reduced chi^2 value simply because it has more flexibility to fit the data.

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