Given a dataset is it possible to decide which curve best fits the data?

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I have a dataset containing weights and heights of 1000 people. I used simple linear regression to arrive at an equation. I wonder why cannot I try to fit an ellipse or a parabola or some curve. Is there is a theory which helps to decide whether a curve or a straight line fits a data.

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It depends on whether you have a model that the data should fit.

In your case, for example, if the bodies are all of similar shape, you might expect that $weight \sim a\ height^3 $ for some $a$.

Therefore $\log weight \sim \log a +3\,\log height $ so a linear plot of log weight vs. log height might be informative. In particular, if this does not show a linear relationship, it might indicate that the proposed relationship does not hold.