📊 Median of Grouped Data
🤔 WHY Do We Need Median for Grouped Data?
Real-Life Importance
Imagine you're a school principal analyzing student performance across different mark ranges. You have data like:
- 📈 Large datasets: 1000+ students' marks
- 📊 Grouped information: Data in intervals (0-10, 10-20, etc.)
- 🎯 Central tendency needed: What's the "middle" performance?
- 📋 Decision making: Resource allocation, remedial classes
📋 HOW TO PLAN: Step-by-Step Strategy
📊 Understand Your Data Type
Identify if your data is in class intervals (grouped) and check if frequencies are given.
📈 Create Cumulative Frequency Table
Build a cumulative frequency column to track running totals.
🎯 Find n/2 (Middle Position)
Calculate half of total observations to locate median position.
🔍 Identify Median Class
Find the class interval containing the median position.
⚡ Apply Median Formula
Use interpolation formula to find exact median value.
⚖️ COMPARISON: Less Than vs Greater Than Types
📉 Less Than Cumulative Frequency
Definition: Shows how many observations are less than the upper boundary of each class.
| Marks | Frequency (f) | Less than (Cumulative) |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 3 | 5 + 3 = 8 |
| 20-30 | 4 | 8 + 4 = 12 |
| 30-40 | 3 | 12 + 3 = 15 |
📈 Greater Than Cumulative Frequency
Definition: Shows how many observations are greater than or equal to the lower boundary of each class.
| Marks | Frequency (f) | Greater than (Cumulative) |
|---|---|---|
| 0-10 | 5 | 53 (total) |
| 10-20 | 3 | 53 - 5 = 48 |
| 20-30 | 4 | 48 - 3 = 45 |
| 30-40 | 3 | 45 - 4 = 41 |
📉 Less Than Type
- ✅ Direction: Ascending order
- ✅ Formula: Add frequencies progressively
- ✅ Usage: More common in textbooks
- ✅ Graph: Increases from left to right
📈 Greater Than Type
- ✅ Direction: Descending order
- ✅ Formula: Subtract from total
- ✅ Usage: Alternative representation
- ✅ Graph: Decreases from left to right
📐 THE MEDIAN FORMULA
🎯 Master Formula for Grouped Data
Lower limit of median class
Sum of all frequencies
Frequency before median class
Frequency of median class
Size of class interval
🎯 WORKED EXAMPLE
📚 Student Marks Analysis
Let's find the median marks for 53 students with the following distribution:
| Marks | Number of Students (f) | Cumulative Frequency (cf) |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 3 | 8 |
| 20-30 | 4 | 12 |
| 30-40 | 3 | 15 |
| 40-50 | 3 | 18 |
| 50-60 | 4 | 22 |
| 60-70 | 7 | 29 |
| 70-80 | 9 | 38 |
| 80-90 | 7 | 45 |
| 90-100 | 8 | 53 |
🎯 Find n/2
Total students (n) = 53
n 2 = 53 2 = 26.5
🔍 Identify Median Class
Look for cumulative frequency ≥ 26.5
First cf ≥ 26.5 is 29 (in 60-70 class)
Median Class = 60-70
📊 Extract Values
Lower boundary
Total frequency
CF before median class
Median class frequency
Class width (70-60)
⚡ Apply Formula
Median = l + n 2 - cf f × h
= 60 + 26.5 - 22 7 × 10
= 60 + 4.5 7 × 10
= 60 + 0.643 × 10
= 60 + 6.43
💡 Key Tips for Success
🎯 Common Mistakes to Avoid
- ❌ Forgetting to find n/2
- ❌ Using wrong cumulative frequency
- ❌ Incorrect class width calculation
- ❌ Wrong median class identification
✅ Pro Tips
- ✅ Always double-check cf column
- ✅ Verify median class carefully
- ✅ Use fraction format in exams
- ✅ Show all substitution steps
🎓 Practice Problems
🔥 Challenge Yourself!
Problem 1: Weekly Pocket Money
| Pocket Money (₹) | Number of Students |
|---|---|
| 0-50 | 8 |
| 50-100 | 12 |
| 100-150 | 15 |
| 150-200 | 10 |
| 200-250 | 5 |
Find the median pocket money.
Problem 2: Heights of Basketball Players
| Height (cm) | Number of Players |
|---|---|
| 150-160 | 3 |
| 160-170 | 7 |
| 170-180 | 12 |
| 180-190 | 8 |
| 190-200 | 4 |
Calculate the median height of the team.
🔗 NCERT/AP-TS Syllabus Connection
📖 NCERT Chapter 14
- ✓ Measures of Central Tendency
- ✓ Median for Grouped Data
- ✓ Cumulative Frequency
- ✓ Applications in Real Life
🎯 AP/TS Board Focus
- ✓ Formula Application (5 marks)
- ✓ Step-by-step Solutions
- ✓ Interpretation of Results
- ✓ Graph-based Problems
📝 Exam Pattern
- ✓ 2-3 marks: Direct formula
- ✓ 4-5 marks: Word problems
- ✓ 6-8 marks: Multiple parts
- ✓ Graph construction
🚀 Advanced Concepts
📈 Understanding Linear Interpolation
The median formula uses linear interpolation - assuming data is evenly distributed within each class interval.
🔍 Visual Understanding
22 students
7 students
24 students
We need 4.5 more students from the median class to reach position 26.5
📊 Ogive (Cumulative Frequency Curve)
An alternative graphical method to find median using cumulative frequency curves.
📈 Steps for Ogive Method
- Plot cumulative frequency vs upper class boundaries
- Draw a smooth curve connecting the points
- From n/2 on y-axis, draw horizontal line to curve
- From intersection point, drop vertical line to x-axis
- Reading on x-axis = Median value
🎯 Advantage: Visual representation helps understand data distribution
📊 Accuracy: Both formula and ogive methods give same result
🎯 Real-World Applications
🏥 Healthcare
Analyzing patient wait times, treatment durations, and recovery periods in hospitals.
💼 Business
Employee salary analysis, sales performance, and customer satisfaction ratings.
🎓 Education
Student performance analysis, grade distribution, and standardized test scores.
🌍 Environment
Temperature distributions, rainfall patterns, and pollution level analysis.
📚 Quick Reference Card
🔢 Formula Components
- l: Lower boundary of median class
- n: Total number of observations
- cf: Cumulative frequency before median class
- f: Frequency of median class
- h: Class interval width
⚡ Quick Steps
- Find n/2
- Locate median class (cf ≥ n/2)
- Identify l, cf, f, h
- Apply formula
- Calculate and interpret
🏆 Master Formula
Memorize this formula - it's your key to success! 🔑
🎉 Conclusion
🏅 You've Mastered Median of Grouped Data!
You now understand the WHY, HOW, and WHEN of finding median for grouped data. This powerful statistical tool will help you analyze real-world data distributions and make informed decisions.
Solve more problems daily
Use in real-life scenarios
Master the formula
Keep practicing and you'll excel in your Grade 10 statistics! 💪