Scatterplots & Correlation (r) – AP Statistics

Learning Objectives

Pearson's $r$: Measures the strength and direction of a purely LINEAR relationship. Fails completely on curved data!

Bounds $[-1, 1]$: $r=1$ is a perfect uphill straight line. $r=-1$ is a perfect downhill line. $r=0$ means absolutely no linear pattern (a shapeless cloud).

$R^2$ (Coefficient of Determination): Measures the % of variance in $y$ completely explained by the linear relationship with $x$. An $R^2$ of 0.81 means 81% of changes in $y$ are predictable by $x$.

Tags

CorrelationScatterR-SquaredBivariate

       PEARSON CORRELATION (r)

       $r = \frac{1}{n-1}\sum \left(\frac{x_i-\bar{x}}{s_x}\right)\left(\frac{y_i-\bar{y}}{s_y}\right)$

CLICK GRAPH TO ADD POINTS MANUALLY

Generate Random Cloud

Target Correlation $r$ = 0.8

SAMPLE METRICS ($n$: 0)

Correlation ($r$): 0.00

Determination ($R^2$): 0.0%

LSRL DIRECTION

Eq $\hat{y}$: ----

The Shape of $r$: Play with the target $r$ slider. At $r = 0$, the cloud is a shapeless circle, meaning $X$ tells you nothing about $y$. At $r = 0.99$, it maps perfectly to a tight diagonal ribbon. Note how negative $r$ flips the slope downwards!