Words like "strong" and "weak" are vague. The correlation coefficient $r$ puts one number on the direction and strength of a linear pattern in a scatterplot. It is built from z-scores, using each variable's standard deviation.
Key idea
$$r = \dfrac{1}{n - 1}\sum z_x z_y,$$ the average product of the z-scores. Always $-1 \le r \le 1$. The sign gives the direction; values near $\pm 1$ mean the dots hug a line; values near 0 mean no linear pattern. $r$ has no units.
Why it works
Split the scatterplot at the point of means $(\bar{x}, \bar{y})$. A dot in the upper right has $z_x > 0$ and $z_y > 0$, so its product is positive; a dot in the lower left has two negatives, also a positive product. Dots in the other two corners give negative products. If most dots sit in the upper-right and lower-left corners, the products add to a large positive number: a positive association. Because z-scores have no units, changing inches to centimeters cannot change $r$, and swapping x and y cannot either.
Worked example: Computing r by hand
Find $r$ for the points $(1, 3), (2, 5), (3, 4), (4, 7), (5, 6)$.
- Means: $\bar{x} = 3$, $\bar{y} = 5$. Deviations: x: $-2, -1, 0, 1, 2$; y: $-2, 0, -1, 2, 1$.
- Both variables have $\sum(\text{dev})^2 = 10$, so $s_x = s_y = \sqrt{10/4} \approx 1.58$, and $z_x z_y = \dfrac{(\text{x dev})(\text{y dev})}{2.5}$.
- Products of deviations: $4, 0, 0, 2, 2$, so $z_x z_y$: $1.6, 0, 0, 0.8, 0.8$.
- $$r = \dfrac{1.6 + 0 + 0 + 0.8 + 0.8}{4} = \dfrac{3.2}{4} = 0.8$$
Answer: $r = 0.8$: a fairly strong positive linear association.
Worked example: What r does and does not say
For $x = 1, 2, \ldots, 10$ and $y = x^2$, software gives $r \approx 0.97$. Does that mean the relationship is linear?
- The points lie exactly on a curve, $y = x^2$, with no scatter at all.
- $r$ is high because the curve bends only gently over this range, so a line comes close.
- A high $r$ does not prove the form is linear; you must look at the plot (and, soon, the residuals).
Answer: No. The relationship is a perfect curve; r only measures how close it is to a line.
Common mistake
Two classic misreadings. (1) Reading $r$ as the slope: $r = 0.8$ does not mean y rises 0.8 per unit of x; slope depends on units, $r$ does not. (2) Calling a strong curved pattern "no relationship" because $r \approx 0$; a U-shaped pattern can have $r$ near 0. Check by looking at the scatterplot every time.