Math 115: Introduction to Statistics – 8. Associations and Causation

Where We’ve Been

So far, we’ve examined variables one at a time:

Frequency tables and bar plots for a single categorical variable
Summary statistics and histograms for a single numerical variable
Hypothesis tests and CIs for a single proportion

Today’s question: how do we summarize and visualize the relationship between two categorical variables — and what does it mean to say they’re associated?

Association and Independence

When we examine two variables together:

Associated: there is a relationship
- Knowing the value of one variable tells you something about the other
Independent: there is no relationship
- Knowing one variable tells you nothing about the other

Our goal: decide whether two categorical variables are associated.

Comic Characters

15,126 characters from DC and Marvel comics
Two key categorical variables:
- Alignment: good, bad, neutral
- Identity type: secret, public, no dual, unknown
Research question: is alignment associated with identity type?

Contingency Tables

	Coffee	Tea	Total
Morning	25	15	40
Night	35	25	60
Total	60	40	100

Example: 100 students surveyed about drink preference (coffee/tea) and morning/night habit.

A contingency table displays two categorical variables together.

Each cell holds the count for that combination of values

Row totals: counts for each row category

Column totals: counts for each column category

Grand total: the sample size — often called \(n\)

Before we tackle the fifteen-thousand-character comics table, let’s get comfortable with the structure on something smaller. One hundred students were surveyed about two things: their drink preference, coffee or tea, and whether they’re a morning or a night person. Those joint counts — students broken down by both questions at once — are exactly what a contingency table is built to display. Each cell holds the count for one specific combination — the highlighted cell shows that twenty-five of the hundred students prefer coffee and are morning people. The row totals on the right tell us how many cases fall into each row category — forty morning people, sixty night people. The column totals along the bottom do the same for columns — sixty coffee drinkers, forty tea drinkers. And the grand total in the bottom-right is just the sample size — one hundred. You’ll also see this object called a two-way table or a cross-tabulation, same idea with different names.

Comics Contingency Table

align	No Dual	Public	Secret	Unknown	Total
Bad	453	2,106	4,352	7	6,918
Good	640	2,905	2,430	0	5,975
Neutral	377	946	908	2	2,233
Total	1,470	5,957	7,690	9	15,126

Raw counts alone don’t reveal patterns clearly — the row totals are very different.

Conditional Proportions

Conditional proportion: the proportion within a row or column — calculated relative to that group’s total, not the grand total.

Formula (conditioned on rows): (count in cell) / (row total)

\[\begin{array}{c} \text{Proportion of} \\ \text{bad characters with} \\ \text{secret identities} \end{array} = \frac{\text{Bad characters with secret identity}}{\text{Total bad characters}}\]

Conditional proportions reveal patterns that raw counts hide.

Comics Conditional Proportions

align	No Dual	Public	Secret	Unknown
Bad	0.065	0.304	0.629	0.001
Good	0.107	0.486	0.407	0.000
Neutral	0.169	0.424	0.407	0.001

Key finding: Bad characters are more likely to have secret identities (62.9%) than good characters (40.7%) — the rows differ, suggesting alignment and identity type are associated.

Caffeine and Sleep

A campus survey of 200 students asked about caffeine intake and typical sleep duration:

	Under 7 hrs sleep	7+ hrs sleep	Total
High caffeine	75	45	120
Low caffeine	20	60	80
Total	95	105	200

What proportion of high-caffeine students sleep under 7 hours? \(75 / 120 \approx\) 0.625 — about 62.5% of high-caffeine students sleep under 7 hours.
What proportion of low-caffeine students sleep under 7 hours? \(20 / 80 =\) 0.25 — 25% of low-caffeine students sleep under 7 hours.
Do caffeine intake and sleep duration appear associated? Yes. The proportion sleeping under 7 hours is much higher in the high-caffeine group (62.5%) than the low-caffeine group (25%). Knowing one tells you something about the other.

Your turn. A campus survey of two hundred students asked about caffeine intake, classified as high or low, and typical nightly sleep duration, classified as under seven hours or seven plus. Work through all three questions yourself, then advance the slide when you are ready to see the answers. The first question asks for the conditional proportion of high-caffeine students who sleep under seven hours. Out of the one hundred twenty high-caffeine students in the table, seventy-five sleep under seven hours. So seventy-five over one hundred twenty is zero point six two five — sixty-two point five percent of high-caffeine students sleep under seven. The second question swaps the row — same outcome, low-caffeine group. Of the eighty low-caffeine students, twenty sleep under seven hours. Twenty over eighty is exactly zero point two five — twenty-five percent. The third question asks whether caffeine intake and sleep duration appear associated. Yes — sixty-two point five percent of high-caffeine students sleep under seven hours, compared with only twenty-five percent in the low-caffeine group. The two conditional proportions are very different, so knowing caffeine intake really does change what we’d predict about sleep duration — and that’s exactly what association means.

Visualizing Two Categorical Variables

Three bar-plot variants — same comics data, three views:

Dodged

Compare counts across groups

Stacked

Show composition of each group

Standardized

Compare proportions within groups

Standardized is the workhorse for spotting associations — every group on the same vertical scale.

Three bar-plot variants come up when both variables are categorical — same data each time, three different choices about what to put on the y-axis. The dodged version puts every combination in its own bar, side by side — for the comics that’s twelve bars, the three alignments times the four identity types. The y-axis is raw count, and the bad-with-secret bar is the tallest. The downside is that smaller groups like neutral look short next to the bad bars, which can make their internal pattern harder to see even when it’s just as interesting. The stacked version pools each alignment into a single bar and slices it by identity type. Total bar height still equals the group’s total count, so bad is tall and neutral is short. The slicing is informative within a bar, but comparing one identity type across alignments is hard because the slice starts at a different height in each bar. The standardized version stacks the same slices, then rescales every bar to height one. Each slice is now a conditional proportion within that alignment, and every alignment sits on the same zero-to-one scale regardless of how many characters were in it. That last view is the one we’ll lean on whenever the question is are these two variables associated — it shows the conditional proportions we just computed in table form, but visually and side-by-side.

Comics Standardized Bar Plot

Standardized bar plot of identity proportions by alignment

The Secret bands aren’t the same size across alignments — visual evidence of an association.

Summary

Summarizing two categorical variables:

Contingency table: counts for each combination
Conditional proportions: proportions within a row (or column) — reveal patterns hidden by raw counts
Standardized bar plot: visualize the conditional proportions side-by-side

Key concepts:

Associated: the conditional proportions differ across groups
Independent: the conditional proportions are (essentially) the same

Up next (8b): when we can and can’t conclude that one variable causes changes in the other.

Three tools came out of this deck for summarizing two categorical variables together. Contingency tables hold the raw counts. Conditional proportions rescale each row to its own total so the rows can be compared cleanly. Standardized bar plots show those same proportions visually, side-by-side. And the working definition: two variables are associated in our sample when conditional proportions differ across groups, independent when they’re essentially the same. In the comics, alignment and identity type pass that test in the sample — bad and good characters have very different secret-identity proportions. Whether that sample-level difference reflects a real population-level association is a formal inference question we’ll pick up later in the course. And even if the population-level association is real, spotting it in the data doesn’t tell us why it’s there. Whether alignment actually causes the difference in identity type, or whether something else is driving the pattern, is what eight B picks up in the next deck.

References

Introduction to Modern Statistics (2e) textbook by Mine Çetinkaya-Rundel and Johanna Hardin
Sections 1.1, 1.2, 4.1, 4.2, 4.3