Unattended

Pl. III
Explorable

Pl. III · Explorable · Statistics, seen 3/3

Statistics, seen · 3 of 3

Least squares, by hand

The best-fit line is the one that makes the squared vertical misses as small as possible. Push the points around and you can feel which ones it listens to.

§1 · Try it

The vermilion line is the least-squares fit. The dashed teal lines are the residuals: how far each point sits above or below the line. Turn on the squared residuals to see what the fit minimises, which is the total area of those squares.

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Fig. 1 · Drag the points Click empty space to add one. Keyboard: Tab to a point, then arrows to move it (Shift for big steps), Delete to remove it. Circled points have high leverage.
Line
ŷ = 1.13 + 0.85·x
Correlation r
0.72
R²
0.52
Sum of squared residuals
26.6
Points
12

§2 · Notice

Three things to notice

  1. The line always passes through the middle of the data, the point (x̄, ȳ). Drag any point and the line pivots, but it never leaves that centre.
  2. Points far to the left or right pull hardest. Press Add a far-out point and the whole line swings towards it. Such points have high leverage: they sit far from x̄, so the line can't avoid them. Points near the middle can move a lot and change little.
  3. r measures how tightly the points hug a line, not how steep it is. A shallow line through a tight cloud has a high r. A steep line through a loose cloud has a low one. R², the square of r, is the share of the variation in y that the line accounts for.

§3 · Play

Guess the correlation

Each cloud has 40 points and an exact, hidden correlation between −1 and +1. Slide to your guess and check it. It is harder than it looks.

Fig. 2 · Round 1 40 points, one hidden correlation.
−1 · falls0 · no pattern+1 · rises

No guesses yet · rounds never run out