Correlation

A scatterplot shows the direction, form and strength of a relationship, but "strong" and "fairly strong" are hard to compare from one plot to the next. For a linear pattern, one number does the job: the correlation, written $r$.

What $r$ tells you

• The sign gives the direction: $r > 0$ for a positive association, $r < 0$ for a negative one.
• The size gives the strength of the linear pattern: near 1 or $-1$ when points hug a line, near 0 when there is no linear pattern.
• $r$ is always between $-1$ and 1. It equals 1 or $-1$ only when every point lies exactly on a line.

The figure shows four data sets. Rough guide: a size above about 0.8 is strong, 0.5 to 0.8 moderate, below 0.5 weak. But look at the fourth plot: the points follow an arch almost perfectly, yet $r = 0.04$. Correlation measures how well a line fits, nothing else.

Where the number comes from

The formula uses z-scores, which you met with one variable. Standardize each $x$ and each $y$, multiply each pair, and average the products (dividing by $n - 1$, as for a standard deviation):

$$r = \frac{1}{n - 1}\sum z_x z_y, \qquad z_x = \frac{x - \bar{x}}{s_x},\quad z_y = \frac{y - \bar{y}}{s_y}.$$

Why does this work? Draw lines at $\bar{x}$ and $\bar{y}$. A point above average on both variables has two positive z-scores, so a positive product; a point below average on both has two negative z-scores, again a positive product. Points in the other two corners give negative products. If most points sit in the upper right and lower left, the products add up to a positive $r$.

Interpreting $r$ in context

For the teacher's 8 students, technology gives $r = 0.951$ between hours studied and quiz score. Write it like this: "There is a strong, positive, linear association between hours studied and quiz score for these students." Do not write "95% of students ..." or "scores rise 0.95 points per hour". $r$ is not a percent and has no units.

Properties to remember

To see the outlier effect: add a ninth student who studied 6 hours but scored 40, and $r$ falls from 0.95 to about 0.14.

Correlation is not causation

A strong correlation in observational data does not mean one variable causes the other. Across cities, the number of fire trucks sent to a fire is strongly correlated with the damage, because bigger fires bring more trucks and do more damage. A third variable linked to both, here the size of the fire, can create a correlation. Only a randomized experiment supports a claim of cause.