Bar Charts: Simple, Composite, and Dual
Bar charts display data using rectangular bars where the length or height corresponds to the frequency or value of each category. They are ideal for comparing discrete categories.
A simple bar chart has one bar per category. For example, if a survey shows the number of students preferring different sports: Football (15), Basketball (10), Tennis (5), a bar chart would have three bars with heights 15, 10, and 5 respectively.
Composite (stacked) bar charts show sub-categories within each main category by stacking bars vertically. For example, if the 15 football fans are split into 9 boys and 6 girls, the football bar is divided into two stacked parts: 9 and 6 units tall, coloured differently to distinguish groups. This allows comparison of both total and subgroup sizes within categories at a glance, but care must be taken to read the total height for overall values and individual segments for subgroups separately to avoid confusion.
- Simple bar chart: one bar per category, height = frequency.
- Composite bar chart: bars divided into stacked segments representing subgroups.
- Dual (side-by-side) bar chart: bars for subgroups placed side-by-side for each category to compare subgroups directly.
Pie Charts: Representing Proportions of a Whole
Pie charts display data as sectors of a circle, where each sector's angle is proportional to the category's share of the total. They are useful for showing how a whole is divided into parts.
To construct a pie chart: first find the total frequency, then calculate each category's fraction of the total, multiply by 360° to find the sector angle.
For example, if 40 students choose subjects: Maths (16), English (12), Science (8), History (4), total = 40. Maths sector angle = (16/40) × 360° = 144°. English = (12/40) × 360° = 108°, and so on. Use a protractor to draw each sector accurately from the centre of the circle, starting at 0° and moving clockwise or anticlockwise consistently.
- Calculate total frequency.
- Find fraction of total for each category.
- Multiply fraction by 360° to get sector angle.
- Draw sectors with a protractor starting from a fixed line.
- Label sectors clearly with category and percentage or frequency.
Pictograms: Using Images to Represent Data
Pictograms represent data quantities using repeated images or symbols, where each image corresponds to a fixed number of items.
For example, if one apple symbol represents 5 students, and 20 students like apples, draw 4 apple symbols.
When quantities are not multiples of the image value, partial symbols or a key explaining the scale can be used. Pictograms are visually engaging but less precise than numerical charts, so they are best for approximate comparisons.
- Decide a scale (e.g., 1 symbol = 5 items).
- Draw whole symbols for multiples of the scale.
- Use partial symbols or keys for non-multiples.
- Label pictogram with scale and category names.
- Ensure symbols are evenly spaced and clear.
Stem-and-Leaf Diagrams: Displaying Ordered Data with a Key
Stem-and-leaf diagrams organise data to show distribution while retaining original values. The 'stem' represents leading digits, and the 'leaf' represents trailing digits.
For example, data: 23, 25, 27, 31, 33, 35. Stems: 2, 3. Leaves for stem 2: 3, 5, 7; for stem 3: 1, 3, 5.
A key explains how to read the diagram, e.g., '3 | 5 means 35'. This helps interpret the data correctly and is essential in exams to avoid ambiguity.
- Order data from smallest to largest.
- Separate each number into stem and leaf parts.
- List stems in a vertical column.
- Write leaves in ascending order beside each stem.
- Include a key to explain stem and leaf values.
Basic Frequency Distributions: Summarising Data
A frequency distribution lists data values or class intervals alongside their frequencies, showing how often each value or range occurs.
For example, test scores: 70, 75, 80, 70, 85, 75, 80, 80. Frequency distribution: 70 (2), 75 (2), 80 (3), 85 (1).
Frequency distributions simplify data analysis and are the basis for constructing other charts like bar charts and histograms.
- List all distinct data values or intervals.
- Count how many times each occurs (frequency).
- Present data in a table with two columns: value/interval and frequency.
- Use frequency data to construct charts or calculate statistics.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Bar Chart Construction | Draw bars with heights proportional to frequencies. | Sports preferences: Football (15), Basketball (10). | Check bar heights match data exactly for full marks. |
| Pie Chart Construction | Calculate sector angles: (frequency/total) × 360°. | Maths (16/40) × 360° = 144° sector. | Use protractor carefully; label sectors clearly. |
| Pictogram Construction | Choose scale (e.g., 1 symbol = 5 items), draw symbols accordingly. | 20 students liking apples = 4 apple symbols. | State scale clearly; partial symbols if needed. |
| Stem-and-Leaf Diagram | Split data into stems and leaves; order leaves; include key. | Data: 23, 25, 27 → 2 | 3 5 7; key: 2 | 3 = 23. | Always include key; order leaves ascending. |
| Frequency Distribution | List values or intervals with frequencies. | Scores: 70(2), 75(2), 80(3). | Present neat table; frequencies must sum to total data points. |
Active recall
Quick check
1. How do you calculate the angle for a sector in a pie chart?
Divide the category frequency by the total frequency, then multiply by 360°. For example, if frequency is 12 and total is 40, angle = (12/40) × 360° = 108°.
2. What does each segment in a composite bar chart represent?
Each segment represents the frequency of a subgroup within the main category; the total bar height is the sum of all segments.
3. Why is a key important in a stem-and-leaf diagram?
It explains how to interpret the stems and leaves, ensuring the original data values are understood correctly.
4. If one symbol in a pictogram represents 5 items, how many symbols represent 18 items?
3 full symbols (representing 15 items) and a partial symbol for the remaining 3 items, or clearly state the partial symbol's meaning.
5. What information does a frequency distribution table provide?
It shows how often each data value or interval occurs in the dataset, summarising the data for analysis.