632 words

3 min read

Visual companion

Python

Type and operator map

Python Week 1: the first filter for runtime behavior

Open gallery Related surface

View

Revision summary

What this note is really saying

Short form

# Statistics I · Week 4 — Association and correlation Two variables together: **tables** for categorical pairs, **scatterplots** and **$r$** for numerical pairs. Association is not causation.

Statistics I · Week 4 — Association and correlation

Two variables together: tables for categorical pairs, scatterplots and $r$ for numerical pairs. Association is not causation.

Week map

Two variables → contingency table → marginal totals → scatterplot features → correlation

r

→ strength vs direction → limitations of

r

Contingency table notation

Contingency table → cross-classified counts → rows = one variable, columns = another.
Cell → count in one row category AND one column category.
Marginal total → row or column sum → not joint.
Joint → specific pair; marginal → one variable alone.

Mini-table (transport × gender, counts):

	Bus	Train	Total
F	30	20	50
M	25	25	50
Total	55	45	100

Cell (F, Train) = 20. Marginal female total = 50.

Scatterplot language

For numerical

x

and

y

Direction → positive (upward), negative (downward), none.
Form → linear, curved, clusters.
Strength → tight cloud vs scattered.
Outliers → points off main pattern.

Mini-description: “positive, roughly linear, moderate strength, one high outlier.”

Correlation $r$

$r$ → Pearson correlation coefficient → measures linear association.
Range $-1 \le r \le 1$ .
$r \approx 1$ → tight increasing line.
$r \approx -1$ → tight decreasing line.
$r \approx 0$ → weak linear pattern (curved pattern may still exist).

Trap:

r

is not steepness of slope — standardized measure of linear fit strength/direction.

Properties (linear world):

Sign of $r$ matches slope of best-fit line.
$r$ unchanged if you add constant to all $x$ or scale $x$ by positive constant (linear transform).

Association vs causation

Pattern may be due to:

Direct cause
Confounding third variable
Coincidence

Ice cream sales and drowning correlate — weather confounds both.

Pattern families

Easy — Read contingency cell

Identify joint count.
Compute marginal from table.
Percent of total in one cell: cell/n.

Medium — Describe scatter

State direction, form, strength in words.
Match sign of $r$ to scatter direction.
Spot outlier effect on $r$ .

Hard — Interpret $r$ and limits

Near ±1 vs near 0 interpretation.
Strong nonlinear pattern with $r$ near 0.
Do not infer causation from $r$ .
Categorical coded as numbers — $r$ may mislead.

Worked mini-examples

Example 1 — Cell percent.

From table above: percent female bus riders = 30/100 = 30% of all, or 30/50 = 60% of females.

Example 2 — Scatter words.

Points rise left to right tightly → positive, linear, strong.

Example 3 — $r$ sign.

r = -0.82

→ strong negative linear association.

Example 4 — Nonlinear trap.

Parabolic cloud around 0 →

r

might be near 0 though

y

clearly depends on

x

Example 5 — Outlier.

Most points tight positive line; one far low point pulls

r

down.

Traps

Causation from correlation.
$r$ for categorical disguised as numbers (zip codes).
Confusing strength with steep slope.
Ignoring outliers dominating $r$ .
Contingency cell vs marginal confusion.

Diagnostic (try yourself)

In a 2×3 contingency table, what does a single cell count represent?
Describe in words a scatterplot that is negative, linear, and weak.
If $r = 0.15$ between study hours and exam score, is linear association strong or weak? Does high score cause more hours?
Table: exercise (Low/High) × health (Poor/Good). Low-Poor cell = 40, total $n=200$ . What percent of all cases are Low and Poor?
A U-shaped scatter has $r \approx 0$ . Can $y$ still depend on $x$ ? Explain briefly.