Diagrams and Tables
Section: Statistics | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Tally Charts and Frequency Tables
A tally chart records data using tally marks, grouped in 5s (|||| with a diagonal line through it = 5). A frequency table shows the total count for each category or value.
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Question: 15 students report the number of pets they own: 0,1,2,1,0,3,1,2,2,1,0,1,2,3,1. Construct a frequency table.
- Step 1: Count each value: 0 appears 3 times, 1 appears 6 times, 2 appears 4 times, 3 appears 2 times
- Step 2: Check the total: 3 + 6 + 4 + 2 = 15 ✓
- Answer: 0→3, 1→6, 2→4, 3→2
Two-Way Tables
A two-way table organises data by two categories at once, with row and column totals to check consistency.
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Question: 50 students are asked whether they prefer Football or Basketball. 15 boys prefer Football, 10 boys prefer Basketball, 8 girls prefer Football, 17 girls prefer Basketball. Complete the table with totals.
- Step 1: Boys total = 15 + 10 = 25. Girls total = 8 + 17 = 25
- Step 2: Football total = 15 + 8 = 23. Basketball total = 10 + 17 = 27
- Check: 25 + 25 = 50, and 23 + 27 = 50 ✓
- Answer: Boys: 25, Girls: 25, Football: 23, Basketball: 27, Grand total: 50
Venn Diagrams
A Venn diagram shows how sets overlap. Items in both categories go in the overlapping region.
A Venn diagram: start with the overlap, then work outward
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Question: In a class of 30 students, 18 play football, 12 play basketball, and 6 play both. Find how many play only football, only basketball, and neither.
- Step 1: Only football = 18 − 6 = 12
- Step 2: Only basketball = 12 − 6 = 6
- Step 3: At least one sport = 12 + 6 + 6 = 24. Neither = 30 − 24
- Answer: Only football = 12, only basketball = 6, neither = 6
Carroll Diagrams
A Carroll diagram sorts data using two criteria, each split into yes/no, forming a grid of four cells.
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Question: Sort 4, 7, 10, 15, 18, 21, 24, 9 into a Carroll diagram using "Even" and "Multiple of 3".
- Even and multiple of 3: 18, 24
- Even and not a multiple of 3: 4, 10
- Not even and multiple of 3: 15, 21, 9
- Not even and not a multiple of 3: 7
Choosing the Right Representation
- Tally/frequency table: counting occurrences of categorical or discrete data.
- Two-way table: comparing two categorical variables at once.
- Venn diagram: showing overlaps between sets ("and"/"or"/"both"/"neither").
- Carroll diagram: sorting data by two yes/no criteria without overlapping regions.
Real-World Applications
These representations organise data across many fields:
- Surveys and market research: two-way tables comparing preferences by demographic.
- School data: Venn diagrams showing subject enrolment overlaps.
- Classification: Carroll diagrams sorting organisms or objects by two properties.
- Sports statistics: frequency tables of goals scored per match.
- Business reporting: organising sales data by category and region.
Exam Tips
Common Mistakes
MistakeMiscounting tally marks, especially when grouping in 5s
Fixgroup tally marks in sets of 5 (four verticals plus one diagonal) to make counting easier
MistakeNot checking that row and column totals in a two-way table match the grand total
Fixalways add up rows and columns separately, then check both match the overall total
MistakePlacing a "both" item in only one circle of a Venn diagram
Fixitems in both categories go in the OVERLAP region, not in either circle alone
MistakeForgetting the "neither" region outside both circles in a Venn diagram
Fixremember to account for items that belong to neither set
MistakeApplying only one criterion when using a Carroll diagram
Fixa Carroll diagram sorts by TWO criteria at once - check both before placing an item
For Exams
- Show tally marks clearly, grouped in 5s.
- Double-check two-way table totals (rows, columns, and grand total).
- In Venn diagrams, start with the overlap value first, then work outward.
- For Carroll diagrams, check each item against both criteria before placing it.
- Choose your representation based on what the question asks you to compare or show.
Interactive revision notes, videos and practice questions load below.