Module 8 — Bivariate I: contingency tables and scatter plots

Data Description | Part III — Two variables and synthesis

Modified

October 8, 2026

The business question. Every univariate indicator so far described one variable at a time. The real question behind “The Wandering Fork”’s trading is whether two variables move together — does warmer weather really bring more revenue? Do promotions work better on sunny days? This module introduces the two graphical tools for answering that: the contingency table (for two qualitative-or-grouped variables) and the scatter plot (for two quantitative variables).

Learning objectives

By the end of this module, you will be able to:

Build and read a contingency table crossing two variables.
Compute row and column marginal distributions from a contingency table.
Build a scatter plot of two quantitative variables and describe the pattern it shows.
Compute a conditional distribution — the distribution of one variable restricted to a single modality of the other — and use it to compare sub-groups fairly.
Distinguish “a visible pattern” from “a proven relationship” — formalised numerically in Module 9.

Time plan

Block	Minutes	What you do
Theory	60	Contingency table, marginal distributions, conditional distributions, scatter plot
Excel lab	60	`COUNTIFS`, PivotTable (two fields), XY scatter chart
Exercises	60	Three by-hand and four Excel exercises
Total	180

Theory

Contingency table

Definition — contingency table

A contingency table (or cross-tabulation) crosses the modalities of two variables observed on the same individuals: cell \((i,j)\) holds the count \(n_{ij}\) of individuals with modality \(i\) of the first variable and modality \(j\) of the second. The row margin \(n_{i\cdot} = \sum_j n_{ij}\) and column margin \(n_{\cdot j} = \sum_i n_{ij}\) recover each variable’s own (univariate) frequency table from the same data.

In plain words: a contingency table is a two-dimensional frequency table. The margins are the row and column totals — they give you back the one-variable frequency tables you already know.

Conditional distribution

Definition — conditional distribution

A conditional distribution is the distribution of one variable restricted to a single modality of the other — one row (or one column), rescaled to its own row (or column) total instead of the grand total \(n\). It answers “given this category, how is the other variable distributed?”, which the raw counts alone cannot, since row totals differ in size.

In plain words: to compare Sunny vs Rainy days fairly, don’t compare raw counts (Sunny has more days total). Instead, compute percentages within each weather type — that’s the conditional distribution.

Scatter plot

Definition — scatter plot

A scatter plot places one point per individual, using one quantitative variable for the \(x\)-coordinate and another for the \(y\)-coordinate. It is the standard first look at a possible relationship between two quantitative variables, before computing any numerical summary.

In plain words: each day becomes a dot at (temperature, revenue). The cloud of dots shows the pattern — upward trend, downward trend, curved, or no pattern at all.

Worked example

Contingency table — `weather` × `daily_revenue_eur` class

From “The Wandering Fork”’s Harbour 30-day subset (wandering-fork.xlsx, sheet harbour_30d):

Weather Revenue class	[400,500[	[500,600[	[600,700[	[700,800[	[800,900[	Row total
Sunny	0	3	6	3	1	13
Cloudy	3	2	2	3	2	12
Rainy	3	1	0	1	0	5
Column total	6	6	8	7	3	30

Step 1 — check the margins. The row totals (13, 12, 5) match Module 2’s weather frequency table exactly, and the column totals (6, 6, 8, 7, 3) match Module 3’s daily_revenue_eur class table exactly — both are recovered as marginal distributions of the same 30 rows, confirming the table was built correctly.

Step 2 — read the pattern. All 6 days in the lowest revenue class [400,500[ were either Cloudy or Rainy — zero Sunny days earned under €500. Conversely, Sunny days account for 10 of the 18 days earning €600 or more, including 6 of the 8 days in [600,700[. This is a first, purely visual hint that better weather is associated with higher revenue.

Step 3 — conditional distribution. Restricting to Sunny days only (row total 13) and rescaling each cell by 13 instead of 30 gives the conditional distribution of revenue class given Sunny: \(0\%, 23.1\%, 46.2\%, 23.1\%, 7.7\%\) for the five classes (rounded; they sum to \(100\%\) before rounding) — compare this to the unconditional column percentages from Module 3 (20.0%, 20.0%, 26.7%, 23.3%, 10.0%): Sunny days are visibly shifted toward the higher revenue classes, confirming the pattern from Step 2 with actual percentages instead of raw counts.

Scatter plot — `temperature_c` vs `daily_revenue_eur`

Plotting temperature_c (x-axis) against daily_revenue_eur (y-axis) for all 30 Harbour days shows points trending upward and to the right: as temperature rises, revenue tends to rise too, though the points do not fall on a perfect line — there is visible scatter around the trend.

Using Excel

The sheet harbour_30d of wandering-fork.xlsx has the data. The complete list of function names, with their French equivalents, is on the Excel functions page.

Concept	Excel function / steps
Contingency table	Select the two columns → Insert → PivotTable, one variable in Rows, the other in Columns, `Count` in Values
Row/column margins	The PivotTable’s own “Grand Total” row and column
Conditional distribution	In the PivotTable, right-click Values → Show Values As → % of Row Total (or Column Total)
Scatter plot	Select the two quantitative columns → Insert → Chart → Scatter (X Y)
Two-way `COUNTIFS`	`=COUNTIFS(range1, criterion1, range2, criterion2, …)` for cell counts

For the full table (120 days per franchise) use the Excel Table tblFork and add the franchise as one more condition, for example =COUNTIFS(tblFork[franchise],"Harbour",tblFork[weather],"Sunny",tblFork[daily_revenue_eur],">=600",tblFork[daily_revenue_eur],"<700").

Proof / derivation

Derivation — marginal totals recover the univariate frequency table (click to expand)

Let \(n_{ij}\) be the contingency table’s cell counts, with row modalities \(i=1,\dots,r\) and column modalities \(j=1,\dots,c\). Every one of the \(n\) individuals is counted in exactly one cell, so summing across all columns for a fixed row \(i\) recovers that row modality’s total univariate count: \[ n_{i\cdot} = \sum_{j=1}^{c} n_{ij}. \] For “The Wandering Fork”, row \(i=\text{Sunny}\): \(0+3+6+3+1 = 13\), matching Module 2’s Sunny count exactly — this is not a coincidence but a direct consequence of the contingency table partitioning the same 30 rows both by weather and, independently, by daily_revenue_eur class.

Visual intuition

A contingency table is a two-dimensional histogram: instead of one row of class counts (Module 3), we now have a grid of counts, and reading down a single column or across a single row projects the two-dimensional picture back onto one axis — exactly recovering the one-variable-at-a-time view from Modules 2–3. A scatter plot’s “upward-and-to-the-right” shape is the visual signature of a positive relationship; Module 9 replaces “shape” with a single number, the correlation coefficient.

Interactive demo

Important

TODO: an interactive contingency table builder (drag categories, see conditional distributions update) would be a good Shinylive demo for this module.

Exercises

Exercise 1 (by hand) — Reading a contingency table (click for solution)

Out of the 12 Cloudy days, what proportion earned in the top two revenue classes ([700,800[ or [800,900[)?

Solution. Cloudy row: \(3+2=5\) days in the top two classes out of 12 Cloudy days total, i.e. \(5/12 \approx 41.7\%\).

Exercise 2 (by hand) — Column margin check (click for solution)

Verify that the column total for [600,700[ (8) is consistent with Module 3’s class frequency table.

Solution. Column [600,700[: \(6\,(\text{Sunny}) + 2\,(\text{Cloudy}) + 0\,(\text{Rainy}) = 8\), matching Module 3’s \(n_{[600,700[}=8\) exactly.

Exercise 3 (by hand) — Describing a scatter plot (click for solution)

Without computing any number, how would you describe the scatter plot of temperature_c vs. daily_revenue_eur — direction, and is the relationship perfectly linear?

Solution. The direction is positive (points trend upward to the right). The relationship is not perfectly linear — there is visible scatter of points around any trend line one might draw, meaning other factors besides temperature also influence daily revenue.

Exercise 4 (Excel) — Build the contingency table for Campus (click for solution)

In wandering-fork.xlsx, filter franchise to “Campus” and build a PivotTable crossing weather (Rows) with daily_revenue_eur class (Columns, grouped in €100 bins). What are the row totals? Do they match Module 2’s Campus weather frequencies?

Solution. Campus 120-day row totals: Sunny 50, Cloudy 48, Rainy 22 (total 120). These match the Campus weather frequencies from Module 2 (Sunny 41.7%, Cloudy 40.0%, Rainy 18.3%).

Exercise 5 (Excel) — Conditional distribution: revenue given promo (click for solution)

Using the full 240-row tblFork, build a PivotTable with promo_active (Rows) and daily_revenue_eur class (Columns, €100 bins). Show values as % of Row Total. Compare the conditional distributions for “Yes” vs “No” promo. Is revenue higher when a promo is active?

Solution. There are 26 promo days and 214 non-promo days. Summing the row percentages: below €200, 15.4% (Yes) vs 19.2% (No); €400 or more, 53.8% vs 48.1%; €600 or more, 23.1% vs 23.4%. Promo days have slightly less mass in the lowest classes, but the top of the two distributions is almost identical. The difference is small and rests on only 26 days — and it is descriptive, not causal (other factors may drive both promo and revenue).

Exercise 6 (Excel) — Scatter plot: foot traffic vs revenue (click for solution)

Create a scatter plot of foot_traffic (x) vs daily_revenue_eur (y) for Harbour 120 days. Describe the direction, form, and strength of the pattern.

Solution. Strong positive linear trend: more foot traffic clearly associates with higher revenue. The points cluster tightly around an upward line — much tighter than temperature vs revenue. This visually confirms the \(r \ge 0.80\) correlation from the dataset design.

Exercise 7 (Excel) — Harbour vs Campus: weather and revenue (click for solution)

For each franchise (120 days each), compute the conditional share of days with daily_revenue_eur of €500 or more given each weather category. Is the weather–revenue association equally strong at both trucks?

Solution. For Harbour and Sunny:

=COUNTIFS(tblFork[franchise],"Harbour",tblFork[weather],"Sunny",
          tblFork[daily_revenue_eur],">=500")
 /COUNTIFS(tblFork[franchise],"Harbour",tblFork[weather],"Sunny")

Weather	Harbour	Campus
Sunny	25/53 = 47.2%	19/50 = 38.0%
Cloudy	11/42 = 26.2%	18/48 = 37.5%
Rainy	2/25 = 8.0%	6/22 = 27.3%

At Harbour the share falls sharply from Sunny to Rainy (47.2% to 8.0%): weather is strongly associated with revenue. At Campus the three shares are much closer (38.0%, 37.5%, 27.3%): the association is weaker, so other factors (such as nearby events) matter more there.

Common mistakes

Watch out

Using raw counts to compare rows with different totals. Always use conditional distributions (% of row/column total) for fair comparison.
Confusing marginal and conditional distributions. Marginal = overall (ignoring the other variable); conditional = given a specific value of the other variable.
Reading a scatter plot as proof of causation. A pattern only shows association; Module 9 quantifies the linear association, but neither proves causation.
Forgetting to group continuous variables for contingency tables. daily_revenue_eur must be binned (e.g., €100 classes) before crossing with weather — PivotTable grouping does this automatically.
Using a line chart instead of a scatter plot for two quantitative variables. Line charts imply a sequence (time); scatter plots show the relationship between two variables measured on the same individuals.

Source	Where	What it adds	Time	Verified
OpenStax IBS 2e	§13.1 The Correlation Coefficient r (book pages 536–538)	Correlation concept, scatter plots	~2 pages	checked against the PDF contents (T046)
OpenStax IBS 2e	§13.3 Linear Equations (book pages 539–542)	Linear relationship basics	~3 pages	checked against the PDF contents (T046)
OpenStax IBS 2e	§3.4 Contingency Tables and Probability Trees (book pages 153–164)	Contingency tables, conditional distributions (probability framing — skim)	~11 pages	checked against the PDF contents (T046)
Khan Academy	Unit Analyzing categorical data in the Statistics and probability course; look for lessons on two-way tables and conditional distributions	Short videos and practice on contingency tables	~30–45 min (estimate)	link check pending (T048)
Khan Academy	Unit Exploring bivariate numerical data in the Statistics and probability course; look for lessons on scatter plots and correlation	Short videos and practice on scatter plots	~30–45 min (estimate)	link check pending (T048)