Undergrad Small Reduced Chi Squared interpretation

[Jakub]

Hello everyone,
I would be happy if someone explained the small reduced chi squared value to me. I have fitted a set of measured data with an exponential function, which I need for some sw calculations. The fit seams great, the origin sw also provides the reduced chi squared, but it is very small in this case. I thought it was supposed to be around 1 in the case of good fit. See the picture.
Thanks in advance for any help.

 

[Orodruin]

Where are your error bars on the data points? You cannot appropriately compute a chi square function without the correct error bars.

 

[Jakub]

Where are your error bars on the data points? You cannot appropriately compute a chi square function without the correct error bars.

The Origin SW does it all ... just by looking at the fitted plot, I can't understand the small reduced chi sq

 

[Orodruin]

The Origin SW does it all ... just by looking at the fitted plot, I can't understand the small reduced chi sq
View attachment 223453

This is generally a bad excuse. If you want to understand what is going on you need to understand what goes on inside the black box. If you have not provided the software with a set of errors, it will likely just assume that the errors in each data point is some default value (like one). In that case you cannot interpret the reduced chi square statistically. The measure of how good your fit is will have no statistical meaning unless you provide the software with the error bars in the data points.

 

[Jakub]

This is generally a bad excuse. If you want to understand what is going on you need to understand what goes on inside the black box. If you have not provided the software with a set of errors, it will likely just assume that the errors in each data point is some default value (like one). In that case you cannot interpret the reduced chi square statistically. The measure of how good your fit is will have no statistical meaning unless you provide the software with the error bars in the data points.

Thanks for the explanation. I am working with tapered optical fibers. For a sw simulation I need to approximate them with a function. My idea was that the SW would do the best fit possible, and the error would be the fit minus the actual measured value (input value). I thought this is where was the reduced chi sq calculated from.

 

[Orodruin]

The error is something related to your measurement. Your instrument will typically have some intrinsic precision. This is the error that should go into the analysis. If you have (intending to do so or not) put the measurement errors to one, what you have is indeed something that fits the prediction way better than expected.

Fun fact: As a teacher you can check if your students are ”massaging” the data in a lab by looking to see if their data fits the prediction too well.