Construct two-way frequency tables and use them to estimate conditional probabilities and test independence

Learning goal: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide whether events are independent and to approximate conditional probabilities.

Two categories per object. Suppose each student in a random sample is asked two questions: which grade are you in, and which of mathematics, science and English is your favorite subject? Each student is then classified by two categorical variables at once. A two-way frequency table records how many students fall in each combination: rows for one variable, columns for the other, and a count in every cell. The row totals, column totals and grand total are written in the margins. Because each student is in exactly one cell, the cell counts add up to the grand total, and the table is a complete picture of the sample.

The data set. The following sample is hypothetical, invented for this lesson; it is not a survey of any real school. A random sample of 200 students from a high school gave these counts.

Grade 9: mathematics 20, science 15, English 15 (row total 50). Grade 10: mathematics 18, science 24, English 18 (row total 60). Grade 11: mathematics 16, science 14, English 20 (row total 50). Grade 12: mathematics 10, science 12, English 18 (row total 40). Column totals: mathematics 64, science 65, English 71; grand total 200.

To construct such a table from raw data you tally each response into its cell, then add across for row totals, down for column totals, and check that both sets of totals add to the same grand total. Here 50 + 60 + 50 + 40 = 200 and 64 + 65 + 71 = 200, so the table is consistent.

The table as a sample space. Choose one of the 200 students at random, each equally likely. The sample space is the set of 200 students, and every cell, row and column is an event. "Tenth grader" is the row event with 60 outcomes, probability 60/200 = 0.3. "Favors science" is the column event with 65 outcomes, probability 65/200 = 0.325. "Tenth grader and favors science" is the single cell with 24 outcomes, probability 24/200 = 0.12. This is exactly the subset language of the first lesson, with the table doing the listing for us.

Conditional probabilities from the table. A conditional probability restricts the sample space to one row or one column. The probability that a randomly selected student favors science given that the student is in tenth grade is the fraction of the tenth-grade row that lies in the science column: P(science | grade 10) = 24/60 = 0.4. The denominator is the row total, because the row is the given event. Doing the same for the other subjects in that row: P(mathematics | grade 10) = 18/60 = 0.3 and P(English | grade 10) = 18/60 = 0.3; the three add to 1 because every tenth grader chose exactly one subject.

Now compare across grades. P(science | grade 9) = 15/50 = 0.3, P(science | grade 11) = 14/50 = 0.28, P(science | grade 12) = 12/40 = 0.3. The tenth graders in this sample favor science at a noticeably higher rate, 0.4, than the other grades, which sit near 0.3. Reading down a column instead of across a row gives the reversed conditional: P(grade 10 | science) = 24/65 ≈ 0.369, the fraction of science-favoring students who are in tenth grade. As always, the two conditionals have different denominators and answer different questions.

Why these probabilities are approximations. The table describes a random sample of 200, not the whole school. The value 0.4 is an exact probability for the model "pick one of these 200 students"; it is only an estimate of the probability that a randomly selected student from the whole school favors science given that they are in tenth grade. A different sample of 200 would give somewhat different counts. The standard's word is approximate, and the honest phrasing is "we estimate P(science | grade 10) to be about 0.4 from this sample".

Testing independence in the table. The events "grade 10" and "favors science" would be independent if P(grade 10 and science) equaled P(grade 10) × P(science). (For the two variables, grade and favorite subject, to be independent, the product rule would have to hold for every one of the twelve grade-and-subject cells; a single cell that fails is enough to show the variables are dependent.) From the table, P(grade 10 and science) = 24/200 = 0.12, while P(grade 10) × P(science) = 0.3 × 0.325 = 0.0975. These are not equal, so within the sample the events are dependent. Equivalently, P(science | grade 10) = 0.4 is not the same as P(science) = 0.325. Under independence every row of the table would show the same conditional distribution across the three subjects; this table does not.

A caution goes with this. Because the counts are sample estimates, small differences between P(A | B) and P(A) do not prove dependence in the population; some gap is expected from sampling alone. The comparison in the table is evidence about the sample. Whether it is strong enough evidence about the school is a question for statistical inference, which asks how large a gap sampling variation alone would typically produce. In this unit we say "the table shows dependence in the sample" and stop there.

A fully worked construction. A class collects 40 hypothetical responses recording whether each student has a part-time job and whether they take a bus to school. The raw tallies: job and bus 6; job and no bus 10; no job and bus 14; no job and no bus 10. Rows: job (16), no job (24). Columns: bus (20), no bus (20). Grand total 40. P(bus | job) = 6/16 = 0.375; P(bus | no job) = 14/24 ≈ 0.583; P(bus) = 20/40 = 0.5. Product test: P(job and bus) = 6/40 = 0.15, P(job)P(bus) = 0.4 × 0.5 = 0.2. Not equal; in this class of 40, having a job and taking the bus are dependent, and students with jobs take the bus less often. The comparison uses the row totals as denominators, never the grand total, for the conditionals.

The misconception. Students frequently compute P(science | grade 10) as 24/200 or as 24/65, taking the wrong total. The given event decides the denominator: given grade 10 means divide by the grade-10 row total, 60. Say the given event aloud, find its total in the margin, and divide the cell by that.

Worked row and column reading: From the 200-student table, P(English | grade 12) is the fraction of the grade-12 row in the English column: 18/40 = 0.45. P(grade 12 | English) is the fraction of the English column in the grade-12 row: 18/71 ≈ 0.254. Same cell, different denominators: 40 twelfth graders versus 71 English-favoring students.

Worked independence check for another cell: Is "grade 9" independent of "favors mathematics" in the sample? P(grade 9 and math) = 20/200 = 0.10. P(grade 9) = 50/200 = 0.25 and P(math) = 64/200 = 0.32, product 0.08. Since 0.10 is not 0.08, the pair is dependent in the sample: P(math | grade 9) = 20/50 = 0.4 is above P(math) = 0.32. Ninth graders in this sample favor mathematics more often than the sample as a whole.

Worked construction from raw data: Twenty hypothetical responses to "own a bicycle?" and "walk to school?" are tallied: bike and walk 3, bike and no walk 9, no bike and walk 5, no bike and no walk 3. Table rows bike 12, no bike 8; columns walk 8, no walk 12; total 20. P(walk | bike) = 3/12 = 0.25 and P(walk | no bike) = 5/8 = 0.625. In this small sample the two conditionals differ a lot, but with 20 responses a new sample could easily change them; the table gives an estimate, not a fixed population value.

Represent and record: Copy the 200-student table with all margins. Beside it write the four conditional probabilities P(science | grade) for grades 9 to 12, each as cell over row total, and the reversed P(grade 10 | subject) for each subject as cell over column total. Circle the denominator used in each and state in one sentence why the denominator of a conditional probability is always the count of the given event, which for these conditionals is a row or column total rather than the grand total (the grand total would be the denominator only if the given event were the whole sample space).

Independent performance: Collect data from at least 30 people (classmates, family, a survey the class designs) on two categorical questions, for example grade level and favorite subject among mathematics, science and English. Construct the two-way frequency table with margins and check that row and column totals agree. Estimate the probability of each subject given each grade and compare across grades; estimate one reversed conditional. Test independence of one grade-and-subject pair with the product rule and with the conditional comparison. Success criteria: the table adds up in both directions; every conditional uses the correct margin; every probability is labeled as an estimate from the sample; the independence verdict is stated for the sample only, with a sentence acknowledging sampling variation.