Scatter Plots & Correlation

Section 1 of 2

Scatter Plots

Bivariate data

Univariate = one item; bivariate (paired) = two items together (e.g. temperature and sales). A scatter plot shows each pair as a point $(x,y)$.

Temperature and ice-cream sales: $(15,8),(17,10),(21,15),(22,17),(25,19),(20,16),(19,20),(17,21)$.

Worked example — a curved pattern

The pairs $x: 1,2,3,4,5,6$ with $y: 11,15,17,17,15,11$ are plotted. Do they show a straight-line (linear) relationship?

The points rise then fall — a curve, not a line

So there is no linear correlation

no linear correlation (it’s a curve)

Section 2 of 2

Correlation

Direction & strength

Positive: both rise. Negative: one rises as the other falls. None: no pattern. The nearer the points to a line, the stronger.

The coefficient r

$r$ measures the linear relationship, always between $-1$ and $+1$. $r=1$ perfect positive, $r=-1$ perfect negative, $r=0$ none. Closer to $\pm1$ = stronger.

Worked example — read r

Rainfall vs sunshine: the points are loosely scattered and drift slightly down. From $\{0.6,\ 0.1,\ -0.1,\ -0.6\}$, which $r$ best fits?

Slightly down → negative; very loose → weak

$r = -0.1$ (weak negative)

You try

The temperature/sales scatter rises clearly with points close to a line. From $\{0.8,\ 0.1,\ -0.9\}$, which $r$ fits, and describe the correlation.

Rising and tight → positive and strong.

$r = 0.8$ — strong positive

$r = 0.8$, strong positive

That’s Class 8.

Scatter plots for paired data, and correlation — direction, strength and the coefficient $r$. Class 9: the normal distribution.