2.C Correlation and the Scatterplot

A scatterplot plots pairs of scores, one variable on each axis, so every dot is one person or case measured twice. When the dots form a tight line, the two variables are strongly related; when they scatter into a shapeless cloud, they barely relate at all. The correlation coefficient r boils that whole cloud down to one number. The sign tells you direction: positive means both variables rise together, negative means one rises as the other falls. The size, from 0 to 1 ignoring sign, tells you how tightly the dots hug a straight line. An r of 0 is a round, directionless blob. As r moves toward -1 or +1, the cloud squeezes into a thinner and thinner diagonal line, and at exactly -1 or +1 every dot sits perfectly on that line.

A common trap is reading r = 0.8 as 'the relationship holds 80 percent of the time.' Correlation does not work that way. Instead square r: r-squared gives the proportion of variance in one variable statistically explained by the other. For r = 0.8, r-squared is 0.64, so 64 percent of the variance is explained and 36 percent is left over, due to other factors or noise. A single extreme outlier can distort r sharply, dragging it toward zero or inflating it, which is why researchers always look at the scatterplot itself, not just the number.

Correlation also underlies reliability and validity. A reliable test gives similar scores on repeated measurement, checked by correlating two sets of scores; a valid test correlates with the outcome it claims to predict. Neither reliability nor validity, nor correlation itself, ever implies that one variable causes the other.