Q.The paired values $(x, y)$ of 10 units are given below, where $x \in \{10, 20, 30\}$ and $y \in \{5, 10, 15\}$: $(10,5),\ (10,10),\ (20,10),\ (20,15),\ (30,5),\ (30,15),\ (20,10),\ (10,5),\ (30,15),\ (20,5)$ Construct a bivariate frequency table with $x$ along the rows and $y$ along the columns.

Tally each pair into the cell where its $x$-row meets its $y$-column, then total.

Step 1 — Sort the pairs by $x$-value.

• $x=10$: $(10,5), (10,10), (10,5)$ → $y=5$: 2, $y=10$: 1, $y=15$: 0.
• $x=20$: $(20,10), (20,15), (20,10), (20,5)$ → $y=5$: 1, $y=10$: 2, $y=15$: 1.
• $x=30$: $(30,5), (30,15), (30,15)$ → $y=5$: 1, $y=10$: 0, $y=15$: 2.

Step 2 — Enter the frequencies and totals.

Independent check. Row totals $3+4+3 = 10$; column totals $4+3+3 = 10$; both equal the 10 pairs given — the table is consistent.

Swapping the row and column value of a pair, or losing count of a repeated pair, are the usual errors — pre-counting the pairs per $x$-value (here 3, 4, 3) fixes the row totals and lets each row's cells be checked against its total.