What Two-Way Tables Show — Rows, Columns, and Totals
A two-way table organises data into rows and columns, each representing a different category. Every cell at the intersection shows a count for that combination. A row total sums across one row; a column total sums down one column; the grand total is the overall count (bottom-right cell).
Row total = sum of all cells in that row. Column total = sum of all cells in that column. Grand total = sum of all row totals = sum of all column totals.
Example 1 — Reading a two-way table
Step 1
Identify what the rows represent (Male / Female) and what the columns represent (Car / Bus / Walk).
Step 2
To find "how many females travel by bus": find the Female row, move across to the Bus column → 20.
Step 3
To find the row total for Male: 18 + 12 + 10 = 40. To verify: column totals 28 + 32 + 20 = 80 = grand total. ✓
Quick check: the sum of all row totals must equal the sum of all column totals, and both must equal the grand total. Always verify this to check your table is correct.
✓ Quick Check 1 — Reading and understanding two-way tables
When cells are missing, use the row and column totals to work out the missing values. The key relationship is: row total − known values in that row = missing value. Similarly for columns. Work systematically — find the cell you have enough information to calculate first.
Missing cell = Row total − (sum of other known cells in that row) OR: Missing cell = Column total − (sum of other known cells in that column)
Example 2 — Finding a missing value using the row total
Year 10 row: Football = 14, Swimming = 8, Row total = 30. Find Tennis.
Step 1
Write the equation: Football + Tennis + Swimming = Row total
Step 2
14 + Tennis + 8 = 30
Step 3
Tennis = 30 − 14 − 8 = 8
Example 3 — Finding a missing value using the column total
Swimming column: Year 10 = 8, Year 11 = 9, Grand total = 60, other col totals = 24 + 19 = 43. Find Swimming column total.
Step 1
Column totals must add to the grand total: 24 + 19 + Swimming = 60
Step 2
Swimming col total = 60 − 24 − 19 = 17
Step 3
Check: 8 + 9 = 17. ✓ Matches.
Strategy: always look for the row or column where only one cell is missing — that is the easiest to calculate. Work outwards from there. Always verify using the grand total at the end.
✓ Quick Check 2 — Completing two-way tables
Question 1 of 10
Interpreting Two-Way Tables — Fractions, Percentages, and Comparisons
Once a table is complete, you can calculate fractions and percentages from any cell. Be careful about the denominator — are you asking "out of the total", "out of that row", or "out of that column"? Reading the question carefully is essential.
Fraction of total: cell ÷ grand total. Fraction of row: cell ÷ row total. Fraction of column: cell ÷ column total. Percentage = fraction × 100.
Example 4 — Calculating a fraction from a two-way table
Step 1
Question: "What fraction of all workers travel by bus?" The relevant cell is Bus column total = 32. The denominator is the grand total = 80.
Step 2
Fraction = \(\frac{32}{80} = \frac{2}{5}\)
Step 3
As a percentage: \(\frac{2}{5} \times 100 = \mathbf{40\%}\) of all workers travel by bus.
Example 5 — Multi-step comparison from a two-way table
Step 1
Question: "Are males or females more likely to travel by bus?" Compare the proportions within each group.
Step 2
Male: 12 out of 40 travel by bus = \(\frac{12}{40} = 30\%\). Female: 20 out of 40 = \(\frac{20}{40} = 50\%\).
Step 3
50% > 30%, so females are more likely to travel by bus than males.
Key phrase to remember: "what fraction of X" — always identify whether X is the row total, column total, or grand total. This tells you what goes on the bottom of the fraction.
A survey asked 80 people about their preferred drink. Use the totals to find the missing values.
Tea
Coffee
Water
Total
Under 30
12
8
30
30 and over
18
22
50
Total
30
18
80
A survey asked 60 people which sport they prefer.
Football
Tennis
Total
Male
12
35
Female
10
25
Total
33
60
90 pupils were asked how they travel to school.
Walk
Bus
Total
Year 7
14
40
Year 8
16
50
Total
90
⚠ Things to Watch Out For
Confusing row total with column total: Row totals sum across each row. Column totals sum down each column. The grand total (bottom-right) is the sum of either.
Using the wrong total as denominator: "What fraction of females travel by car?" — denominator is total FEMALES, not total car travellers. Identify whether you want row % or column %.
Not completing totals first: Fill in row and column totals before finding missing internal cells. Missing values are found by subtraction: total − known values in that row/column.
Reading the wrong cell: Clearly identify the row and column intersection for the cell you need. Misreading one row or column gives a completely different value.
Grand total check: Sum of all row totals = sum of all column totals = grand total. If these don't agree, there's an error somewhere in the table.