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Two-Way Frequency Tables

A free Algebra I lesson from the “Statistics and Data Analysis” unit, with a worked example and practice problems including step-by-step solutions.

A two-way frequency table organizes counts by two categorical variables — rows for one variable, columns for the other. Inside cells are joint frequencies; row and column totals are marginal frequencies. Conditional frequencies divide a joint count by a row or column total to ask 'within this group, what percent...?' In Statistics and Data Analysis, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.

What you'll learn

Why it matters: Survey results, medical-test outcomes (true positive / false negative), and product preference by demographic are all two-way tables.

Worked example

Problem. 100 students were surveyed. 15 are in band and play sports; 10 are in band only; 35 play sports only; 40 do neither. How many play sports total?

  1. Plays sports = (band AND sports) + (sports only).
  2. = 15 + 35 = 50.
  3. Connect the result back to Two-Way Frequency Tables so the method and meaning are both clear.
  4. Check the result against the original representation before writing the final answer.

Answer: 50

Practice problems

1. Out of 100 students: 15 are in band and play sports, 10 in band only, 35 play sports only, 40 do neither. How many are in band?

Show solution
  1. Warm-up: First identify exactly what the question is asking: Out of 100 students: 15 are in band and play sports, 10 in band only, 35 play sports only, 40 do neither. How many are in band?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Band = (band AND sports) + (band only) = 15 + 10 = 25.
  4. Check the result by substituting or estimating: the response should match 25 and make sense in the original problem.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: 25

2. Same survey. How many play sports AND are in band?

Show solution
  1. Warm-up: First identify exactly what the question is asking: Same survey. How many play sports AND are in band?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The joint cell for both is 15.
  4. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: 15

3. Same survey. Total students?

Show solution
  1. Warm-up: First identify exactly what the question is asking: Same survey. Total students?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Sum of all four cells: 15 + 10 + 35 + 40 = 100.
  4. Check the result by substituting or estimating: the response should match 100 and make sense in the original problem.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: 100

4. Same survey. What percent play sports?

Show solution
  1. Core Practice: First identify exactly what the question is asking: Same survey. What percent play sports?
  2. For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
  3. 50 out of 100 = 50%.
  4. Check the result by substituting or estimating: the response should match 50 and make sense in the original problem.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: 50

5. Same survey. What percent are in band?

Show solution
  1. Core Practice: First identify exactly what the question is asking: Same survey. What percent are in band?
  2. For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
  3. 25 out of 100 = 25%.
  4. Check the result by substituting or estimating: the response should match 25 and make sense in the original problem.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: 25

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