2.2 Joint, Marginal and Conditional Relative Frequencies

Same Cell, Different Questions

A café records every purchase in a defined lunch period. Each purchase has one payment type and one takeaway/eat-in label. The complete count table is below.

	Takeaway	Eat in	Total
Card	36	24	60
Cash	12	28	40
Total	48	52	100

A joint relative frequency divides an interior cell by the table’s grand total. Card-and-takeaway is $36/100=0.36$. A marginal relative frequency divides a row or column total by that same grand total. Card purchases are $60/100=0.60$; takeaway purchases are $48/100=0.48$. These describe one variable without restricting the other.

A conditional relative frequency restricts the reference group first. Among card purchases, takeaway is $36/60=0.60$. Among takeaway purchases, card is $36/48=0.75$. The same 36 appears in three calculations, but 0.36, 0.60 and 0.75 answer different questions. Read “among,” “of those who,” or “given” as an instruction to choose the restricted denominator.

The entire row-conditional distribution for card purchases is 60% takeaway and 40% eat-in. The cash row has 30% takeaway and 70% eat-in. Alternatively, conditioning on columns gives 75% card/25% cash among takeaway purchases, and about 46.2% card/53.8% cash among eat-in purchases. Each conditional distribution sums to 100% within its own reference group.

Worked Claim: Payment and Purchase Type

We want to assess whether purchase type is associated with payment type. Compare the takeaway shares within payment groups: 60% for card versus 30% for cash. Those differ by 30 percentage points. The conditional distributions differ, so these data support an association. Comparing raw counts 36 and 12 alone would leave group-size differences mixed into the result.

Now consider the claim, “A larger share of card purchases is takeaway than the share of cash purchases that is takeaway.” The calculated conditional percentages justify it directly: $36/60>12/40$. We can name the conclusion precisely for purchases in this lunch period. It does not claim that using a card causes takeaway orders.

Changing the question can change the conditioning direction. If the claim were “Most takeaway purchases are paid by card,” the correct evidence would be $36/48=75\%$, not $36/60$. Before comparing percentages, write down the population each percentage describes. For evidence of association, compare distributions with the same response categories across levels of the conditioning variable. For a specific contextual claim, use the summary statistics whose denominators answer that claim.

Totals also provide checks. The four joint proportions sum to 1, both marginal distributions sum to 1, and each complete row- or column-conditional distribution sums to 1. A conditional percentage can exceed the corresponding joint percentage because its denominator is smaller; that is not a contradiction.

Try It Yourself

The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.

Original generated practice for the Fall 2026 course.

Calculate Relative Frequencies from a Café Table

Complete drink-choice population for one specified café day

Drink choice	Morning	Afternoon	Total
Tea	18	42	60
Coffee	42	18	60
Total	60	60	120

Every one of the 120 orders is classified as tea or coffee and as morning or afternoon. The counts are exact for this day.

Part a

Construct a complete joint-relative-frequency table, including its marginal row/column totals. Identify tea-and-morning, tea overall, and afternoon overall, stating each denominator.

Your reasoning

Part b

Calculate the complete drink-choice distribution among afternoon orders and the complete time-of-day distribution among tea orders. Explain the restricted reference groups and why their totals are each 100%.

Your reasoning

Worked answer and rubric

Compare and Justify a Café Claim

Complete drink-choice population for one specified café day

Drink choice	Morning	Afternoon	Total
Tea	18	42	60
Coffee	42	18	60
Total	60	60	120

Every one of the 120 orders is classified as tea or coffee and as morning or afternoon. The counts are exact for this day.

Part a

Calculate and compare the complete drink-choice distributions in morning and afternoon. Determine whether time of day and drink choice are associated in these data.

Your reasoning

Part b

Justify or reject the claim “Tea is a larger share of afternoon orders than morning orders.” Use calculated summary statistics, a numerical comparison and a precise contextual conclusion. Explain why equal overall tea/coffee totals do not settle this claim.

Your reasoning

Worked answer and rubric