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Here's the situation:

  • Users have manually drawn a straight line of best fit through a set of data points. I have the equation (y = mx + c) for this line.

  • I have used least-squares regression to determine the optimal line of best fit for the same data.

How can I assess the quality of the user-drawn LOBF? My first thought was just to work out the uncertainty between the two gradients and the two y-intercepts, but that produces dramatic errors when the true value of either the gradient or the y-intercept is close to zero. Any suggestions, please?

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Square error is calculated the same way, no matter how you fit a model (OLS, random forest, deep learning, or guessing like you are doing).

$$ \sum_{i=1}^n\bigg( Y_{true, i}-Y_{predicted, i} \bigg)^2 $$

Divide by the sample size if you want an average.

If you want to automate this task for a computer to do it, calculate the slope and intercept of your guesses line of best fit. Pick two points on your line, $(x_0,y_0)$ and $(x_1,y_1)$.

$$ y-y_0 = \dfrac{ y_1-y_0 }{ x_1-x_0 } \bigg( x-x_0 \bigg) $$

When you do the algebra to solve for $y$ in terms of $x$, you will have the equation for the line you’ve fitted by eyeballing it. You then can use this to make predictions to feed into the square error formula.

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