Lesson 12 — Two-Way Tables & Fairness

Summary

Learners use two-way tables and conditional relative frequencies to explore relationships between categories. They compute joint, marginal, and conditional relative frequencies, learn that an overall rate can hide different rates inside subgroups, and use the table to ask questions of fairness — connecting to early data visualization for justice.

Objectives

• Use two-way tables and conditional relative frequencies to explore relationships between categories, including questions of fairness. (D05.S4.09.02)

Connection

Suppose two schools report very different overall pass rates. You might assume one school is “better.” But a two-way table — arranging the data by two categories at once, like school and prior preparation — can show that the difference comes not from the schools but from who enrolls. The overall number hid a fairer picture. Two-way tables are how we check whether a difference is real or an illusion — and how we ask, honestly, who gets what outcome.

Materials

• Two-way-table sheet
• Worked-example card

Preparation

• Copy the two-way-table sheet and worked-example card.
• Retrieval: from Grades 6–8, recall relative frequency (a count divided by a total) and the idea that association is not causation.

Facilitator note

This lesson is written to the learner (“you”). The skill to land: a two-way table organizes counts by two categories; joint frequencies are the cell counts, marginal frequencies are the row and column totals, and a conditional relative frequency divides a cell by its row (or column) total to reveal the rate within a subgroup. Use a worked example first (S-011): a table of an outcome by two groups, computing joint, marginal, and conditional relative frequencies and reading each in a sentence. The critical-thinking lens: an overall rate can hide different rates inside subgroups — the core of Simpson’s paradox — so compute conditional frequencies before concluding. The ethics and egalitarian lenses: fairness questions live in conditional rates (does group A succeed at the same rate as group B given the same starting point?), and data has been used for justice — W. E. B. Du Bois hand-drew charts for the 1900 Paris Exposition showing the real condition of Black Americans against claims of inferiority, and Florence Nightingale used her statistical charts to argue for hospital reform (S-432, S-433). The global lens: cross-tabulating categories is the same honest act in every census and every study worldwide. The technology & environment lens: a spreadsheet computes conditional frequencies instantly, but the human must decide what the table reveals and what further evidence is still needed; and two-way tables are exactly how people track who has safe drinking water, care, or access — the same honest cross-tabulation maps both fairness and the environment. Emphasize: association is not causation — a table shows a relationship, not a cause; saying which requires more evidence.

Procedure

1. Recall (5 min). From Grades 6–8: what is a relative frequency? What does “association is not causation” mean?
2. Meet the table (8 min). A two-way table sorts counts by two categories. The inside numbers are joint frequencies; the row and column totals are marginal frequencies; a cell divided by its row or column total is a conditional relative frequency.
3. Study the worked example (12 min). A table of “passed or not” by “studied in a group or alone.” Fill the joint and marginal frequencies. Then compute the conditional relative frequency of passing given group study, and given solo study. Read each in a sentence: “Among those who studied in a group, ___% passed.”
4. Compute and interpret (15 min). On the two-way-table sheet, work a new table (two neighborhoods by “has safe drinking water or not,” or two groups by an outcome you choose). Compute joint, marginal, and conditional relative frequencies, and write what each reveals.
5. Ask the fairness question (10 min). Does the overall rate match the rates inside the subgroups? If not, what might the overall number be hiding? What further evidence would you need before saying one group is treated unfairly? (Remember: association is not causation.)
6. Close (5 min). A two-way table turns “one number” into “let me look inside.” That is where fairness questions begin.

Differentiation

• Support: Provide the table pre-filled with counts and ask learners to compute only the conditional relative frequencies and read them aloud.
• Extension: Construct a Simpson’s-paradox example — an overall rate that reverses within each subgroup — and explain, in words, how it happens.

Assessment

• Formative (observation): Can the learner compute a conditional relative frequency and read it in a sentence, and state one fairness question the table raises?
• Portfolio artifact: The completed two-way table with conditional relative frequencies and a written fairness question, kept in the portfolio.

Home connection

Think of a real “two categories” question in your life (two stores by “was the price fair?”; two teams by “did everyone get to play?”). Sketch a two-way table of what you would count and one conditional rate you would want to know.

Resources

• On W. E. B. Du Bois’s data visualization for justice: Battle-Baptiste & Rusert, W. E. B. Du Bois’s Data Portraits (S-432).
• On Florence Nightingale’s statistical charts: Wikipedia, “Florence Nightingale,” https://en.wikipedia.org/wiki/Florence_Nightingale (S-433).
• On real two-category data and fairness: Our World in Data, https://ourworldindata.org/ (S-006).
• On explicit instruction and worked examples: Kirschner, Sweller & Clark (2006), https://doi.org/10.1207/s15326985ep4102_1 (S-011).