How to find error between non-linear plot and data points?

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Is there a formal way to measure the error between some arbitrary points and a non-linear curve in order to minimize it?
 

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The short answer is "yes".
You can regress to a curve in many ways... least-squares is popular.
What's best depends on the curve and what you want to know.
I'm guessing you already know how to regress to a line?