Correlation.

Step 1 of 6

Rising and falling

Until now each reading was one number. Many interesting questions are about two numbers for each item: people’s height and weight, hours of study and marks, a flat’s price and its floor area. Each item is a point $(x, y)$⁠, and together they make a scatter plot.

When the points rise from left to right there is a positive correlation: $y$⁠ tends to grow as $x$⁠ grows. When they fall, a negative correlation. The correlation coefficient $r$⁠, between $-1$⁠ and $1$⁠, sums this up in one number. Its sign is the direction, and its size is the strength: $r = \pm 1$⁠ when every point lies on a straight line, and $r = 0$⁠ when there is no straight-line trend at all.

The points on the board fall. Predict $r$⁠, then the board will work it out.

Step 2 of 6

Tighter

The readings on the board were made from a straight rule, $y = 5 + 0.6(x - 5)$⁠, plus noise: a random miss for each point. Noise sets how far the points stray from the rule. Right now the noise is large, $1.6$⁠, and $r = 0.70$⁠.

Step 3 of 6

Steep or tight

If $r$⁠ did not change with the slope, what exactly does it measure? Compare two clouds of points. The board hides $r$⁠ for now.

Step 4 of 6

Aiming for $r$⁠

$r$⁠ depends on how two things compare: how much $y$⁠ changes along the line, and how far the points stray from it. A link with $|r| \ge 0.7$⁠ is usually called strong, one with $|r|$⁠ between $0.3$⁠ and $0.7$⁠ moderate, and below $0.3$⁠ weak. These are rules of thumb, not laws.

Right now $r = 0.89$⁠. Build a link just as strong, the other way.

Step 5 of 6

Not a straight line

Now the slope is $0$⁠, and $r$⁠ is close to $0$⁠: here there really is no link between $x$⁠ and $y$⁠, just noise about a level line.

But an $r$⁠ close to $0$⁠ does not always mean there is no link.

Step 6 of 6

Not a cause

The big danger in reading a correlation is to think it shows a cause. A strong correlation between $x$⁠ and $y$⁠ can happen because $x$⁠ causes $y$⁠, because $y$⁠ causes $x$⁠, because something else causes both, or by chance. The data alone cannot tell these apart.