Quick Summary
Statistics is the branch of mathematics that involves collecting, organising, presenting, and interpreting data. In Class 8, students learn the three measures of central tendency (mean, median, mode) and how to represent data visually using bar graphs and pie charts.
Key Formulas
| Measure | Formula | Use |
|---|---|---|
| Mean | x̄ = Σx / n | Average value of all observations |
| Median (odd n) | Middle value after sorting | When n is odd: position = (n+1)/2 |
| Median (even n) | Average of two middle values | When n is even: average of n/2 and (n/2 + 1) values |
| Mode | Most frequent value | For data with a clear peak |
| Range | Maximum − Minimum | Measure of spread |
Measures of Central Tendency
1. Mean (Arithmetic Average)
Example: Find the mean of {4, 7, 2, 9, 3, 5, 6}
- Sum = 4 + 7 + 2 + 9 + 3 + 5 + 6 = 36
- n = 7
- Mean = 36 ÷ 7 = 5.14
When to use: When all values are important; affected by extreme values (outliers).
2. Median (Middle Value)
Step 1: Arrange data in ascending (increasing) order. Step 2: Find the middle value.
For odd n:
- Median = value at position (n+1)/2
- Example: {2, 3, 5, 7, 9} → n=5, position = 3 → Median = 5
For even n:
- Median = average of values at positions n/2 and n/2 + 1
- Example: {2, 4, 6, 8} → n=4, positions 2 and 3 → Median = (4+6)/2 = 5
When to use: When data has extreme values (outliers) that would distort the mean.
3. Mode (Most Frequent Value)
Definition: The value that appears most often.
Example 1: {3, 5, 3, 7, 3, 9, 5} → 3 appears 3 times → Mode = 3 Example 2 (bimodal): {2, 4, 4, 6, 6, 8} → 4 and 6 both appear twice → Mode = 4 and 6 Example 3 (no mode): {1, 2, 3, 4, 5} → all appear once → No mode
When to use: For categorical data (colours, names) or when the most common value is needed.
Worked Examples
Problem 1: The marks scored by 8 students: 45, 62, 78, 55, 62, 90, 72, 62 Find mean, median, and mode.
Mean: (45+62+78+55+62+90+72+62) = 526 ÷ 8 = 65.75
Median: Sorted: 45, 55, 62, 62, 62, 72, 78, 90 n=8 (even): average of 4th and 5th = (62+62)/2 = 62
Mode: 62 appears 3 times → Mode = 62
Problem 2: The ages of 7 children: 10, 12, 10, 15, 11, 12, 10 Find mean and mode.
Mean: (10+12+10+15+11+12+10) = 80 ÷ 7 = 11.43 Mode: 10 appears 3 times → Mode = 10
Bar Graphs
A bar graph uses rectangular bars of equal width to represent data.
Steps to draw: 1. Label the x-axis (categories) and y-axis (values/frequency). 2. Choose a suitable scale for the y-axis. 3. Draw bars of equal width but varying height. 4. Give the graph a title.
Types:
- Simple bar graph — one set of data
- Double bar graph — compare two sets of data side by side
Reading a bar graph:
- Tallest bar = highest value
- Shortest bar = lowest value
- To find sum: add all bar heights
Pie Charts (Circle Graphs)
A pie chart shows data as parts of a whole circle (360°).
Formula to find angle for each sector: Angle = (Value / Total) × 360°
Example: Monthly expenses — Food: ₹3000, Rent: ₹2000, Travel: ₹1000, Other: ₹2000. Total = ₹8000.
- Food angle = (3000/8000) × 360° = 135°
- Rent angle = (2000/8000) × 360° = 90°
- Travel angle = (1000/8000) × 360° = 45°
- Other angle = (2000/8000) × 360° = 90°
- Check: 135 + 90 + 45 + 90 = 360° ✓
Steps to draw: 1. Find the angle for each component using the formula. 2. Draw a circle. Mark the centre. 3. Draw sectors using a protractor. 4. Label each sector with category name and percentage.
Reading a pie chart:
- Largest sector = largest share
- Sector angle/360° × 100 = percentage
Important Questions
One Mark
1. What is the formula for mean? 2. What is the mode of {3, 5, 3, 7, 5, 5}? 3. Find the median of {2, 4, 6, 8, 10}. 4. What angle does a sector representing 25% occupy in a pie chart? 5. What is the range of {12, 7, 19, 4, 15}?
Short Answer (3 marks)
1. Find the mean, median, and mode of: 8, 4, 6, 8, 3, 2, 8. 2. In a class of 30 students, the marks scored are: 40, 55, 60, 45, 70, 55, 80, 55, 90, 65, 75, 55. Find the mode and median. 3. Represent this data in a pie chart: Science 40%, Maths 30%, English 20%, Tamil 10%.
Long Answer (5 marks)
1. The daily temperatures (°C) for a week were: 28, 31, 27, 30, 29, 33, 28. Find the mean, median, mode, and range. Which measure best represents the typical temperature? 2. Draw a bar graph for the following data: Number of students in classes 6–10: Class 6 = 45, Class 7 = 52, Class 8 = 48, Class 9 = 55, Class 10 = 50.
Common Mistakes
- Forgetting to sort data before finding the median.
- Using the formula Σx/n without adding all values correctly — double-check your sum.
- In pie charts, forgetting to check that angles sum to 360°.
- Drawing bars of different widths in a bar graph — widths must be equal.
Revision Checklist
- [ ] Mean formula memorised and practised with 5 examples
- [ ] Median for both odd and even n practised
- [ ] Mode identified in data with no mode, one mode, and two modes
- [ ] Bar graph drawn and labelled for a dataset
- [ ] Pie chart angles calculated and drawn for a dataset
- [ ] Textbook exercises completed