Residual Plots

A residual is one subtraction: actual value minus predicted value. For each data point, run it through your model, get a prediction, and subtract. Do that for every point and plot the results against x. If your model captured the real shape of the data, those residuals should look like random scatter around zero, no trend, no curve, no funnel shape. That randomness is the signature of a good fit: whatever is left over is just noise.

If instead the residual plot shows a curve, a U-shape, or any systematic bend, your model is missing structure. That leftover shape is literally the piece of the pattern your equation failed to capture. A line fit to data that actually curves upward will produce residuals that dip negative in the middle and rise positive at both ends, or some similar bow shape, because the line cuts through the curve rather than following it. The residual plot is doing diagnostic work: it shows you the shape of your mistake, which tells you what kind of model to try next, maybe quadratic or exponential instead of linear.

This sits right after fitting regression models to data. Once you have a candidate equation, the residual plot is the check: did I pick the right family of function, not just the right numbers. It also connects forward to comparing linear, quadratic, and exponential fits, since a patterned residual plot is often the clue that pushes you toward a log transformation or a different model type entirely.