I’m trying to get my head around drawing a line of best fit and I keep second‑guessing myself. When I plot a small set of points, I feel like I’m trying to thread a straw through a fuzzy cloud and every time I tilt it a tiny bit, the “best” line seems to change.
Example: say I’ve got points (1, 3), (2, 5), (3, 6), (4, 9). If I sketch a line by eye, sometimes it crosses the y‑axis near 2, other times closer to 1, depending on how I try to “balance” the dots. I’ve heard a few different tips: make the numbers of points above and below roughly equal, or make sure it goes through the mean point (x̄, ȳ), or try to make the total vertical distances small. Which of these is the actual rule I should follow when I’m doing it by hand?
Also, does it matter which variable I put on the x‑axis? If I swap axes, the line I draw looks different, which makes me feel like I’m changing the story. I’m confused because different lines give noticeably different predictions (like for x = 5), and I want a consistent way to decide what “best” really means here. What should I aim for when placing the line by eye?
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3 Responses
The actual rule is ordinary least squares: pick your explanatory variable as x, then draw the line that minimizes the sum of squared vertical residuals (it must pass through the mean point (x̄,ȳ)), so by eye aim for a line through (x̄,ȳ) with ups and downs roughly balanced in size, not just in count. Swapping axes changes the criterion (you’d be minimizing horizontal errors instead), so choose axes by which variable predicts which; nice walkthrough: https://www.khanacademy.org/math/ap-statistics/bivariate-data-ap/least-squares-regression/a/regression-line-review
Your straw-in-a-cloud feeling is spot on-until you decide what “best” means, the cloud keeps shifting! The standard rule for a line of best fit (when predicting y from x) is the least-squares line: the line that makes the sum of squared vertical distances as small as possible. Two handy consequences: it must pass through the mean point (x̄, ȳ), and when you include an intercept, the positive and negative vertical residuals balance to zero in total (not necessarily in count). So, by eye: first plot the centroid (x̄, ȳ), plant your ruler through that point, then tilt it so the tall “springs” above and below look roughly balanced in area. Swapping axes changes the rule (you’d then be minimizing horizontal, not vertical, distances), so yes, the line changes; pick x as the explanatory variable when your goal is predicting y from x. If both variables have similar measurement error and you want a symmetric fit, that’s a different creature (orthogonal/Deming regression).
For your points (1,3), (2,5), (3,6), (4,9), the least-squares line works out cleanly: x̄ = 2.5, ȳ = 5.75; slope b = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² = 9.5/5 = 1.9; intercept a = ȳ − b x̄ = 1. So the line is y = 1.9x + 1, and it indeed crosses the y-axis at 1 and goes straight through (2.5, 5.75). A quick prediction at x = 5 gives y ≈ 10.5. If you’re fitting by hand without arithmetic, anchor at the centroid and tilt to balance the “vertical area” of residuals rather than trying to force equal numbers of points above and below. A nice walkthrough of these ideas is here: https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/regression-line-example
Hope this helps!
The “actual rule” people usually mean is the least‑squares regression line: the line that makes the sum of squared vertical distances to the points as small as possible. Two practical takeaways for drawing it by hand: (1) it must pass through the mean point (x̄, ȳ), and (2) you tilt it so the vertical residuals “balance” in a squared‑error sense-roughly, similar positive and negative areas, with points further from x̄ having more leverage on the tilt. For your data, x̄ = 2.5 and ȳ = 5.75, and the least‑squares slope works out to about 1.9, giving y ≈ 1 + 1.9x; you can see it does go through (2.5, 5.75) and crosses the y‑axis near 1. Heuristics like “equal numbers of points above and below” are okay guides, but the mean‑point rule is exact, and minimizing squared vertical distances (not plain distances) is the criterion. About axes: yes, swapping them changes the line, because standard regression minimizes vertical errors of y on x; if you want to predict y from x, keep x on the horizontal. If you swapped, you’d be fitting x on y, a different task. Hope this helps!