Scatter Plots & Line of Best Fit

A scatter plot shows paired data as points. If the points trend upward, the correlation is positive; downward is negative; no pattern means no correlation.

A line of best fit runs through the middle of the trend, with roughly as many points above as below. Once you have it, you can estimate values you did not measure.

Two cautions worth taking seriously.

Predicting inside the range of your data (interpolation) is reasonably safe. Predicting far outside it (extrapolation) is not — trends rarely continue forever.

And correlation does not prove causation. Ice cream sales and drowning rates rise together, but neither causes the other. Hot weather causes both.

A scatter plot displays bivariate numerical data. The direction, form and strength of the association are described qualitatively, and the correlation coefficient r quantifies linear strength on a scale from −1 to 1, with values near 0 indicating no *linear* relationship — which does not rule out a strong nonlinear one.

The least-squares regression line minimises the sum of squared residuals, and is what a calculator's linear regression returns. Its slope is interpreted as the predicted change in y per unit change in x, and its intercept only carries meaning when x = 0 lies within a sensible range of the data.

Residuals are the practical diagnostic: a residual plot with visible structure indicates that a linear model is the wrong form regardless of how large r is.

The causation caveat is a matter of study design, not statistics. Association can arise from causation in either direction, from a confounding variable, or from coincidence, and only a controlled experiment can distinguish them.