Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Quick Summary
Statistics organises and summarises data. Class 9 covers frequency distribution tables, histograms, frequency polygons, and the three measures of central tendency (mean, median, mode). This is a high-scoring chapter with predictable question patterns.
Key Definitions
- Data: Collection of facts, numbers, or measurements.
- Raw Data: Data collected but not organised.
- Frequency (f): Number of times a value or class appears.
- Class Interval: A range of values grouped together (e.g., 0–10).
- Class Mark (Midpoint): (Upper limit + Lower limit) / 2.
- Cumulative Frequency: Running total of frequencies up to a given class.
- Measures of Central Tendency: Mean, Median, Mode.
Frequency Distribution Table
| Class Interval | Tally | Frequency (f) | Class Mark (x) | fx | Cumulative Frequency |
|---|---|---|---|---|---|
| 0–10 | IIII | 4 | 5 | 20 | 4 |
| 10–20 | IIII II | 7 | 15 | 105 | 11 |
| ... | ... | ... | ... | ... | ... |
Mean
Ungrouped data: Mean = Σx / n
Grouped data: Mean = Σfx / Σf
Assumed Mean method: Mean = A + (Σfd / Σf) where d = x − A
Median
Ungrouped (odd n): Middle value when arranged in order. Ungrouped (even n): Average of the two middle values.
Grouped data: Median = L + [(N/2 − cf) / f] × h
Mode
Ungrouped: Most frequently occurring value.
Grouped data: Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
where f₁ = modal class frequency, f₀ = preceding class frequency, f₂ = succeeding class frequency.
Empirical Relationship
Mode ≈ 3 Median − 2 Mean
Graphical Representations
- Histogram: Bar graph for grouped continuous data (no gaps between bars).
- Frequency Polygon: Connect midpoints of histogram tops; extend to x-axis at both ends.
- Ogive (Cumulative Frequency Curve): Plot cumulative frequency vs upper class boundary.
Important Questions
One Mark
1. Find the mean of 5, 8, 10, 12, 15. 2. What is the class mark of the interval 20–30? 3. Define mode. 4. Find the median of 3, 5, 7, 9, 11. 5. State the empirical relationship between mean, median, and mode.
Short Answer — 3 marks
1. Find the mean of the following grouped data: (10–20: 3), (20–30: 5), (30–40: 7), (40–50: 5). 2. The marks of 10 students are: 72, 65, 80, 58, 75, 65, 90, 70, 65, 85. Find the mode and median. 3. Draw a frequency polygon for the data: (0–5: 4), (5–10: 8), (10–15: 12), (15–20: 6), (20–25: 2).
Long Answer — 5 marks
1. The weekly wages (in ₹) of 30 workers are given in a frequency distribution table (classes 500–600, 600–700, etc.). Find the mean wage using the assumed mean method. 2. For a grouped frequency distribution, construct a cumulative frequency table and draw an ogive. From the ogive, estimate the median.
Common Mistakes
- Using the wrong boundary in the median formula — always use the lower class boundary, not the lower class limit.
- Forgetting to find cf (cumulative frequency) before the median class.
- Reporting mode as the class interval instead of the midpoint in some problems.
- Not connecting the polygon to the x-axis at both ends when drawing frequency polygons.
Revision Checklist
- [ ] Can construct a frequency distribution table from raw data
- [ ] Know and can apply the mean formula for both ungrouped and grouped data
- [ ] Know the median formula for grouped data with all variables
- [ ] Know the mode formula for grouped data
- [ ] Know the empirical formula: Mode = 3 Median − 2 Mean
- [ ] Can draw a histogram and frequency polygon