Statistics — Class 10 Mathematics
"Without data, you are just another person with an opinion." — W. Edwards Deming
1. About the Chapter
Statistics = collecting, organising, analysing, interpreting data.
Class 10 focuses on grouped data — large datasets organised into class intervals.
Three Measures of Central Tendency
- Mean (average)
- Median (middle value)
- Mode (most frequent value)
Why Important
- Election polls and analyses
- Economic data (inflation, GDP)
- Medical research
- Sports statistics
- Climate data
- Business intelligence
2. Grouped Data — Recap
Class Intervals
Data is organised into ranges (classes):
- 0-10
- 10-20
- 20-30
- etc.
Frequency
Number of data items in each class.
Frequency Distribution Table
| Class | Frequency (f) |
|---|---|
| 0-10 | 5 |
| 10-20 | 12 |
| 20-30 | 8 |
| 30-40 | 7 |
Important Terms
- Class mark = (Lower limit + Upper limit) / 2
- Class size = Upper limit − Lower limit (constant)
- Cumulative frequency = running total of frequencies
3. Mean of Grouped Data
Direct Method
Mean = Σ(fᵢxᵢ) / Σfᵢ
where:
- fᵢ = frequency of i-th class
- xᵢ = class mark of i-th class
Example
| Class | f | x (mark) | fx |
|---|---|---|---|
| 0-10 | 5 | 5 | 25 |
| 10-20 | 12 | 15 | 180 |
| 20-30 | 8 | 25 | 200 |
| 30-40 | 7 | 35 | 245 |
| Total | 32 | 650 |
Mean = 650 / 32 = 20.3125
Assumed Mean Method (Easier Calculation)
For large numbers: Mean = a + (Σfᵢdᵢ / Σfᵢ)
where:
- a = assumed mean
- dᵢ = xᵢ − a
This simplifies arithmetic.
Step-Deviation Method
Mean = a + (Σfᵢuᵢ / Σfᵢ) × h
where:
- a = assumed mean
- uᵢ = (xᵢ − a) / h
- h = class size
Useful for clean numbers.
4. Median of Grouped Data
Definition
Median = middle value (when data is arranged in order).
For grouped data, the median lies within a specific class — the MEDIAN CLASS.
Finding Median Class
- Calculate cumulative frequency (CF)
- Find class where CF ≥ N/2 first (where N = total frequency)
- That's the median class.
Formula
Median = l + ((N/2 − cf) / f) × h
where:
- l = lower limit of median class
- N = total frequency
- cf = cumulative frequency of class BEFORE median class
- f = frequency of median class
- h = class size
Example
| Class | f | CF |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 12 | 17 |
| 20-30 | 8 | 25 |
| 30-40 | 7 | 32 |
N = 32, N/2 = 16. First CF ≥ 16 is 17 (in class 10-20).
- l = 10, cf = 5, f = 12, h = 10
- Median = 10 + ((16 − 5) / 12) × 10 = 10 + 110/12 = 10 + 9.17 = 19.17
5. Mode of Grouped Data
Definition
Mode = most frequent value.
For grouped data, the mode is in the MODAL CLASS — class with highest frequency.
Formula
Mode = l + ((f₁ − f₀) / (2f₁ − f₀ − f₂)) × h
where:
- l = lower limit of modal class
- f₁ = frequency of modal class
- f₀ = frequency of class BEFORE modal class
- f₂ = frequency of class AFTER modal class
- h = class size
Example
From the same table:
- Modal class: 10-20 (highest f = 12)
- l = 10, f₁ = 12, f₀ = 5, f₂ = 8, h = 10
- Mode = 10 + ((12 − 5) / (24 − 5 − 8)) × 10
- = 10 + (7 / 11) × 10
- = 10 + 6.36 = 16.36
6. Empirical Formula
For most distributions: Mode = 3 × Median − 2 × Mean
Or: Mean − Mode = 3(Mean − Median)
These are approximate but useful relationships.
Verification (from above example)
- Mean = 20.31, Median = 19.17, Mode = 16.36
- 3 × Median − 2 × Mean = 3(19.17) − 2(20.31) = 57.51 − 40.62 = 16.89 ≈ Mode ✓
7. Ogives (Cumulative Frequency Curves)
What is an Ogive?
A graph of CUMULATIVE FREQUENCY against class boundaries.
Two Types
Less Than Ogive:
- Plot (upper limit of each class, less than CF)
- Curve rises gradually
- Used when 'how many less than this value'
More Than Ogive:
- Plot (lower limit, more than CF)
- Curve falls gradually
- Used when 'how many more than this value'
Finding Median from Ogive
Both ogives intersect at the median (graphical method).
8. Worked Examples
Example 1: Mean
Heights of 20 students (cm):
| Class | f | x | fx |
|---|---|---|---|
| 130-140 | 4 | 135 | 540 |
| 140-150 | 8 | 145 | 1160 |
| 150-160 | 6 | 155 | 930 |
| 160-170 | 2 | 165 | 330 |
| Total | 20 | 2960 |
Mean = 2960 / 20 = 148 cm
Example 2: Median
From Example 1 data:
- CF: 4, 12, 18, 20
- N/2 = 10. First CF ≥ 10 is 12 (in class 140-150).
- Median class: 140-150
- l = 140, cf = 4, f = 8, h = 10
- Median = 140 + ((10 − 4) / 8) × 10 = 140 + 7.5 = 147.5 cm
Example 3: Mode
- Modal class: 140-150 (f = 8 highest)
- l = 140, f₁ = 8, f₀ = 4, f₂ = 6, h = 10
- Mode = 140 + ((8 − 4) / (16 − 4 − 6)) × 10
- = 140 + (4/6) × 10 = 140 + 6.67 = 146.67 cm
9. Common Mistakes
-
Class mark wrong
- Class mark = (lower + upper) / 2. For 10-20, it's 15, not 20.
-
Cumulative frequency confusion
- Just keep adding frequencies.
-
Median class wrong
- Find FIRST CF ≥ N/2. Don't pick last.
-
Empirical formula misuse
- 3 × Median − 2 × Mean = Mode (when needed).
-
Class size assumption
- Class size is the WIDTH (upper − lower), constant across classes.
10. Real-World Applications
Census Data
Indian census every 10 years uses statistics.
Economic Indicators
- Inflation rate, GDP, unemployment — all statistics
- Indian Statistical Service handles national data
Health
- Average heights, weights of children
- Disease prevalence rates
- COVID-19 statistics
Education
- Class average marks
- School performance comparisons
- Indian SSA, NCERT use statistical tools
Sports
- Cricket batting averages
- Football scores
Indian Context
- Indian Statistical Institute (founded 1931) — world-leading
- PM Modi government uses statistics for policy
11. Indian Heritage
Statistical Methods in India
- Aryabhata (5th c.) used statistical methods in astronomy
- C.R. Rao (1920-2023) — world-renowned Indian statistician
- P.C. Mahalanobis (1893-1972) — founder of ISI, Indian planning statistician
12. Conclusion
Statistics is the language of MODERN ANALYSIS:
- Politics (polling)
- Economics (data)
- Science (research)
- Daily life (sports, weather)
Master:
- Three measures: mean, median, mode
- Calculation methods for grouped data
- Ogives for graphical median
- Empirical formula
In Class 11-12, you'll learn variance, standard deviation, regression.
Statistics: the science of making sense of data.
