Paper Together: sit a paper in a room with your friends.See how it works
This page sets out the method for two-way tables in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Two-way tables topic page.
When you see a two-way table with blanks, hunt for a line with exactly one gap.
When there is no table drawn, draw one yourself with the two categories on the sides.
Each cell is the overlap of a row category and a column category.
“Boys who walk” is one cell, not a total.
A row total is the sum of the cells in that row.
The grand total is the sum of the row totals, or of the column totals. Both routes match.
Subtract the known cells in that row from the row total.
Do the same with the column if the row total is the missing fact.
“Boys or girls who walk” might be a whole column.
Read whether the question wants one cell, a total, or a comparison.
A percentage of a row uses the row total as the denominator.
A percentage of everyone uses the grand total.
After you fill a gap, the row sum and the column sum should both work.
If they disagree, the gap is in the wrong place.
Find the row and the column before you copy a number. "Year 7 and bus" is the cell where that row meets that column. The row total to the right includes bus, walk and every other way of travelling. It is not the bus count. The column total underneath includes every year group. It is not the Year 7 count.
The grand total is the one number that is both a row total and a column total. Students sometimes add it into a row as if it were another category. That double-counts everyone. Data cells are inside the table. Totals are on the edge.
Method: 1. Name the row and the column of the cell you need. 2. Read the number where they meet. 3. Do not use a total when the question asks for one category. 4. Treat the corner number as the grand total, not as another category.
A row total is an ordinary addition across that row. Year 7 boys 11 and Year 7 girls 8 would make a Year 7 total of 19. A column is the same idea downwards. Boys in Year 7 and boys in Year 8 add to the boys total. Adding a row when you were asked for a column mixes two different groups.
The grand total is a check. Add the row totals. Add the column totals. Both sums are the grand total, so they match. This is worth doing even when the question does not ask for it. A table that fails the check has a copying error, and every later probability would use the wrong denominator.
Method: 1. Add across for a row total. 2. Add down for a column total. 3. Add the row totals and add the column totals. 4. Those two sums should both equal the grand total.
If a row total is known, the missing cell is that total minus the cells you can see. Year 9 has total 26 and 18 right-handed pupils, so the left-handed cell is 26 - 18 = 8. Use the column as a second route when you can. If both routes are available and they disagree, look again at which total you subtracted from.
Fill the easiest gap first. A gap in the grand total is the sum of the row totals you already trust. A gap in a column total is the grand total minus the other column totals. Once an edge number is filled, it unlocks the cell inside. Write each new number in the table before you move, so you do not subtract a number you have not updated.
Method: 1. Choose a row or a column that contains the gap and a known total. 2. Subtract the known cells from that total. 3. Find the same cell from the other direction if you can. 4. Check the two routes agree, then fill the next empty cell.
A probability from a finished table is a fraction. The numerator is the cell, or the total, that matches the event. The denominator is the group you are choosing from. Chosen at random from everyone, that group is the grand total. Twenty-two students who do both sport and music, out of 60 students, is 22/60, which simplifies to 11/30.
"Given that the student does sport" changes the denominator to the sport row total. The numerator is still the cell inside that row. Using the grand total for a "given that" question, or using a row total for "from the whole school", answers a different question. Underline the group in the sentence before you write the fraction.
Method: 1. Finish every missing count before you divide. 2. Put the matching cell, or its total, on the top of the fraction. 3. Use the group named in the question as the bottom of the fraction. 4. Simplify the fraction by dividing the top and bottom by the same number.
A two-way table has rows, columns and a total for each, with a grand total in the bottom right corner. Every row adds across to its total and every column adds down to its total. The grand total is the sum of the row totals and also the sum of the column totals. Never add a row total and a column total together.
Start by writing in every number the question gives you. Then look for a row or column with only one blank. That blank is the total minus the numbers already there. Fill it in, and look again. Each new number gives you another row or column with one blank. Work like this and you never have to guess. At the end, check that the grand total is the same whether you add across or down.
Method: 1. Copy every given number into its cell. 2. Find a row or column with exactly one blank. 3. Work out the blank by subtracting from the total, and write it in. 4. Repeat until the table is full, then check the grand total both ways.
Tip: Use a pencil and tick each clue as you use it. If a clue is left over at the end, the table is wrong or you missed a cell.
Exam questions often give the facts as sentences: "62 pairs were sold without a discount", "48 pairs were trainers", "20 of the trainers were sold without a discount". Each sentence names a row, a column or one cell. A sentence with "of the" names a single cell inside a row or column, so "20 of the trainers were sold without a discount" is the cell where Trainers meets Without discount.
Decide first which are totals and which are cells. "48 were trainers" is a column total. "20 of the trainers" is a cell inside that column. Write the sentence on the table in the right place before you do any arithmetic. Words like "the rest" mean subtract from the relevant total. Once the cells are placed, use the order from the last section to finish the table.
Method: 1. Read each sentence and decide if it gives a total or a single cell. 2. Write the number in the correct row and column. 3. Use "the rest" to subtract from the matching total. 4. Finish with the one-blank method and check the totals.
To find a probability, use the table as the sample space. For a person chosen from everyone, the denominator is the grand total. For "a girl is chosen" the denominator is the girls total. For "a student who walks is chosen" it is the total of the walk column. The numerator is the cell or total that meets the condition.
Always identify the group the person is chosen from, and write that total as the denominator. A student chosen at random from all 80 members who is a girl and walks has a probability of 14/80, while a girl chosen from the girls total of 25 who walks has a probability of 14/25. Both numerators are the same, but the questions are different. Simplify your fraction at the end.
Method: 1. Decide which group the person is chosen from. 2. Use that total as the denominator. 3. Find the cell that meets the condition for the numerator. 4. Write the fraction and simplify it.
At higher grades the cells are given as percentages or ratios. A club of 72 with boys to girls in the ratio 4 : 5 has 72 ÷ 9 = 8 in one part, so 32 boys and 40 girls. Then "3/8 of the boys play football" gives 3/8 × 32 = 12 boys. Fill in the table at each stage rather than keeping numbers in your head.
Algebra can appear too: cells such as x, x + 6 and 2x - 1 whose sum is the grand total. Write an equation from the sum of the cells, solve it, then substitute back into every cell. Check that the cells add to the total. Two-step probability without replacement uses the table twice: after one boy is chosen, there is one fewer boy and one fewer person in the denominator.
Method: 1. Turn each percentage or ratio into a number and write it in the table. 2. For algebra, add the cells to the total and solve. 3. Substitute back and check every row and column. 4. For two choices without replacement, reduce the numerator and the denominator by 1 for the second choice.
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