Making Predictions from a Line of Best Fit

Scattered data rarely lines up perfectly, but if the points trend together, a single straight line can capture that trend well enough to predict values you never actually measured. The line of best fit is the one that balances all the points at once: it does not pass through most of them, but it makes the overall gaps between the line and the real data as small as possible. Once you have that line's equation, you treat it like any linear function. Plug in an x value, even one between your data points, and read off the predicted y.

To build the equation from two points on the line, find the slope the normal way: change in y over change in x. Then use one point and that slope to write the equation, and plug in whatever x you need. For (20, 1.5) and (40, 3.3), the slope is (3.3 - 1.5)/(40 - 20) = 0.09. Starting from (20, 1.5), the prediction at x = 32 is 1.5 + 0.09(32 - 20) = 2.58. The gap between a real measurement and the line's prediction at that same x is the residual: measured minus predicted. A measured value of 2.0 under a fitted value of 2.4 gives a residual of 2.0 - 2.4 = -0.4, meaning the actual point sits below the line.

This idea connects directly to scatterplot models and to judging how well a line fits: residuals that are small and scattered randomly above and below the line mean a good linear fit, while a clear pattern in the residuals means a line is the wrong shape. Predictions only make sense near the x values the data actually covered, which is why pushing far outside that range is treated separately as a warning sign rather than a reliable use of the model.