Bivariate data record two variables for each observational unit. If the observations are

$$(x_1,y_1),\ldots,(x_n,y_n),$$

a scatterplot places each pair as a point in the coordinate plane.

Reading a scatterplot

A scatterplot can reveal the direction, form and strength of an association.

An upward trend suggests positive association, a downward trend negative association, and a diffuse cloud little obvious association. Curved patterns may be strong even when a straight-line summary is weak.

Outliers, clusters and gaps should be examined before reducing the data to one number.

Sample covariance

A sample analogue of covariance is

$$s_{xy}=\frac1{n-1}\sum_{i=1}^n(x_i-\bar x)(y_i-\bar y).$$

Its sign describes whether observations tend to deviate from their means in the same or opposite directions, but its magnitude depends on measurement units.

Sample correlation

Standardizing by the sample standard deviations gives the sample correlation

$$r=\frac{s_{xy}}{s_xs_y},$$

with

$$-1\le r\le1.$$

Correlation summarizes linear association. A value near zero does not rule out a nonlinear relationship.

Association and causation

An observed association can arise from direct causation, reverse causation, common causes, selection effects or chance. Neither a scatterplot nor a large correlation coefficient by itself establishes that changing one variable would cause the other to change.

Bivariate analysis begins by describing the relationship actually present in the data before any causal interpretation is attempted.