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Raw data is messy. Before you can find an average or draw a chart, you need to organise the data into a table. The two workhorses of GCSE statistics are the frequency table, which records how often each value occurs, and the two-way table, which sorts data by two categories at once. Reading and completing these tables accurately is one of the most reliable sources of marks in the OCR GCSE Mathematics (J560) Statistics strand.
This topic sits mostly in AO1 (filling in and reading tables correctly) with a strong AO2 strand (using totals to work out missing entries and answering questions from the table). OCR command words you will meet here include "Complete the table", "Work out" and "How many …". The skill is careful, systematic arithmetic — get the running totals right and the marks follow.
| Term | Meaning |
|---|---|
| Tally | Marks grouped in fives (the fifth stroke crosses the previous four) used to count quickly |
| Frequency | The number of times a value or category occurs |
| Frequency table | A table listing each value alongside its frequency |
| Two-way table | A table that classifies data by two different variables at once |
| Total (margin) | A row or column sum; the overall total sits in the bottom-right corner |
| Discrete frequency table | A frequency table for separate values such as numbers of pets |
A tally chart is the quickest way to count data by hand. Each item scores one stroke, and every fifth stroke is drawn diagonally across the previous four. This grouping in fives makes the final count fast and reduces miscounts. The frequency is simply the total number of strokes for each row.
For example, the goals scored by a netball team in 20 matches were: 2, 0, 1, 3, 2, 1, 0, 2, 4, 1, 2, 3, 1, 0, 2, 1, 3, 2, 1, 0.
Counting these gives the frequency table:
| Goals | Tally | Frequency |
|---|---|---|
| 0 | four marks | 4 |
| 1 | five-then-one (a group of five and one more) | 6 |
| 2 | five-then-one | 6 |
| 3 | three marks | 3 |
| 4 | one mark | 1 |
| Total | 20 |
Always check the frequencies add to the number of data items — here 4+6+6+3+1=20 ✓. A bar chart of these frequencies makes the shape of the data clear at a glance:
The shoe sizes of 25 pupils are recorded. From the tally, the frequencies are: size 4 → 3, size 5 → 7, size 6 → 8, size 7 → 5, size 8 → 2. Show that the frequencies are consistent with 25 pupils, and state the most common shoe size.
Solution: Total =3+7+8+5+2=25 ✓, which matches the 25 pupils. The most common size is the one with the highest frequency: size 6 (frequency 8).
A frequency table of the number of texts sent by 30 students in an hour is partly complete: 0 texts → 5, 1 text → 9, 2 texts → ?, 3 texts → 4, 4 texts → 2. Work out the missing frequency.
Solution: The frequencies must add to 30. So far 5+9+4+2=20. The missing frequency =30−20=10. So 10 students sent 2 texts.
The table shows the number of siblings of 40 students.
| Siblings | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency | 7 | 15 | 11 | ? | 2 |
(a) Find the missing frequency. (b) How many students have at least 2 siblings?
Solution: (a) Missing frequency =40−(7+15+11+2)=40−35=5. (b) "At least 2" means 2, 3 or 4 siblings: 11+5+2=18 students.
Common error: for "at least 2", forgetting to include the 2-siblings group, or accidentally including the 1-sibling group.
When data is spread over many values — such as heights or times — listing every value is impractical. Instead we group the data into class intervals. Each interval must not overlap, so we use inequality notation. For continuous data a class might be written 150≤h<160, meaning "150 up to but not including 160".
The times, in seconds, taken by 30 students to solve a puzzle are grouped below. Complete the frequency column total and state the class with the most students.
| Time t (seconds) | Frequency |
|---|---|
| 0<t≤10 | 4 |
| 10<t≤20 | 11 |
| 20<t≤30 | 9 |
| 30<t≤40 | 6 |
Solution: Total =4+11+9+6=30 ✓. The class with the most students (the highest frequency, 11) is 10<t≤20.
The masses of 50 parcels are grouped. The frequencies for four of the five classes are 8, 14, 15 and 6, and the fifth class has the remaining parcels. Work out the missing frequency.
Solution: Known total =8+14+15+6=43. Missing frequency =50−43=7 parcels.
A frequency table is more than a list — it is a compact summary you can interrogate. Once it is complete, you can answer a whole family of questions from it: how many items fall above or below a threshold, what fraction lie in a particular class, or which class is the most common. The single most useful habit is to keep a running cumulative total in your head (or in an extra column) as you read down the table. Cumulative totals turn "how many scored fewer than 30?" into a one-step lookup and are the exact tool you will use for medians and quartiles in later lessons. Whenever you complete a frequency table, finish by reading it back: does the bottom total match the number of data items, and do the individual rows make sense in the context of the question?
The grouped table shows the distances, in km, that 48 cyclists rode one weekend.
| Distance d (km) | 0<d≤20 | 20<d≤40 | 40<d≤60 | 60<d≤80 |
|---|---|---|---|---|
| Frequency | 7 | 18 | ? | 9 |
(a) Work out the missing frequency. (b) How many cyclists rode 40 km or less? (c) What fraction rode more than 40 km? Give your answer in its simplest form.
Solution: (a) Missing frequency =48−(7+18+9)=48−34=14. (b) 40 km or less covers the first two classes: 7+18=25 cyclists. (c) More than 40 km covers the last two classes: 14+9=23, so the fraction is 4823 (already in simplest form). As a check, 25+23=48, the whole group ✓.
Common error: double-counting the boundary value. Because the class is written 40<d≤60, a ride of exactly 40 km belongs to the previous class, 20<d≤40, not to "more than 40".
A two-way table classifies data by two variables at once — for example, gender across the top and choice of activity down the side. The clever part is the margins: the right-hand column gives each row total, the bottom row gives each column total, and the bottom-right cell gives the grand total. Because every row and every column must add up correctly, you can fill in missing entries by subtraction.
The two-way table shows whether students walk or take the bus to school, split by year group. Complete the table.
| Walk | Bus | Total | |
|---|---|---|---|
| Year 7 | 18 | 30 | |
| Year 8 | 14 | 26 | |
| Total |
Solution: Work cell by cell, using any row or column that has just one gap.
The completed table is:
| Walk | Bus | Total | |
|---|---|---|---|
| Year 7 | 18 | 12 | 30 |
| Year 8 | 12 | 14 | 26 |
| Total | 30 | 26 | 56 |
Common error: trying to fill a cell from a row and a column at once. Always complete a line that has only one missing value, then use the new figure to unlock the next.
The two-way table shows the lunch choice of 80 students. Complete it.
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