📊 Summary Statistics: Mean, Median & Mode
The Story of the Three Wise Averages
Imagine you’re the coach of a basketball team. After every game, you want to know: “How did we do overall?”
You can’t remember every single score. You need ONE number to tell the whole story.
That’s where Mean, Median, and Mode come in — the three wise friends who summarize your data in different ways!
🎯 What Are Summary Statistics?
Summary statistics are like a photo album’s cover photo. Instead of showing 100 pictures, one image captures the memory.
When you have lots of numbers (like test scores, ages, or prices), summary statistics give you one number that represents them all.
📐 ARITHMETIC MEAN
The “Fair Share” Calculator
What Is It?
The arithmetic mean is what most people call “the average.”
Think of it this way:
🍕 You and 4 friends order pizzas. You get different amounts of slices: 2, 3, 5, 4, and 6 slices.
If you shared everything equally, how many slices would each person get?
Add them up → Divide by how many people
The Formula
Mean = Sum of all values ÷ Number of values
Or written fancy:
Mean = (x₁ + x₂ + x₃ + ... + xₙ) / n
Example: Pizza Slices
Slices: 2, 3, 5, 4, 6
- Add them: 2 + 3 + 5 + 4 + 6 = 20
- Count them: 5 people
- Divide: 20 ÷ 5 = 4
✅ Mean = 4 slices per person
If everyone shared fairly, each person gets 4 slices!
Real-Life Example
Your test scores this month: 70, 80, 85, 90, 75
- Sum: 70 + 80 + 85 + 90 + 75 = 400
- Count: 5 tests
- Mean: 400 ÷ 5 = 80
Your average score is 80! 🎉
⚖️ WEIGHTED MEAN
The “Some Things Matter More” Average
What Is It?
Not everything is equally important!
🎓 Imagine your final grade: Homework counts 20%, but the Final Exam counts 50%. The exam matters more — it has more weight.
The weighted mean gives more importance to things that deserve it.
The Formula
Weighted Mean = (w₁×x₁ + w₂×x₂ + ...) ÷ (w₁ + w₂ + ...)
Where:
- x = each value
- w = the weight (importance) of each value
Example: School Grades
| Assessment | Score | Weight |
|---|---|---|
| Homework | 90 | 20% |
| Midterm | 80 | 30% |
| Final Exam | 70 | 50% |
Step 1: Multiply each score by its weight
- Homework: 90 × 0.20 = 18
- Midterm: 80 × 0.30 = 24
- Final Exam: 70 × 0.50 = 35
Step 2: Add the products 18 + 24 + 35 = 77
Step 3: Divide by total weight 0.20 + 0.30 + 0.50 = 1.00
77 ÷ 1 = 77
✅ Weighted Mean = 77
Notice: Even though you got 90 on homework, your final grade is 77 because the exam (where you scored 70) counted more!
🔗 COMBINED MEAN
The “Merge Two Groups” Average
What Is It?
What if you have two groups with their own averages, and you want to find the average of everyone together?
🏫 Class A has 20 students with average height 140 cm. Class B has 30 students with average height 150 cm. What’s the average height of all 50 students?
You can’t just average the two averages (that would be wrong!). You need the combined mean.
The Formula
Combined Mean = (n₁×Mean₁ + n₂×Mean₂) ÷ (n₁ + n₂)
Where:
- n₁, n₂ = size of each group
- Mean₁, Mean₂ = average of each group
Example: Two Classes
| Class | Students | Average Height |
|---|---|---|
| A | 20 | 140 cm |
| B | 30 | 150 cm |
Step 1: Multiply each mean by its group size
- Class A: 20 × 140 = 2,800
- Class B: 30 × 150 = 4,500
Step 2: Add the totals 2,800 + 4,500 = 7,300
Step 3: Divide by total students 20 + 30 = 50 students
7,300 ÷ 50 = 146 cm
✅ Combined Mean = 146 cm
Why Not Just Average 140 and 150?
(140 + 150) ÷ 2 = 145 cm ❌ WRONG!
Class B has more students, so their average pulls the combined mean closer to 150. That’s why we get 146, not 145!
📍 MEDIAN
The “Middle Kid” of Your Data
What Is It?
Line up all your numbers from smallest to largest. The median is the one standing right in the middle.
🧒 Think of 5 kids standing in height order: Short → Medium-short → MIDDLE → Medium-tall → Tall
The middle kid IS the median!
Finding the Median
Step 1: Sort numbers from smallest to largest
Step 2: Find the middle
- Odd count? → The middle number is the median
- Even count? → Average the two middle numbers
Example 1: Odd Number of Values
Test scores: 85, 70, 90, 75, 80
Step 1: Sort them 70, 75, 80, 85, 90
Step 2: Find middle 5 numbers → Middle is position 3
✅ Median = 80
Example 2: Even Number of Values
Ages: 12, 15, 18, 22
Step 1: Already sorted! 12, 15, 18, 22
Step 2: Two middle numbers Position 2: 15 Position 3: 18
Step 3: Average them (15 + 18) ÷ 2 = 16.5
✅ Median = 16.5
Why Median Matters
The median ignores outliers!
Salaries: 30K, 35K, 40K, 45K, 1,000K (CEO’s salary!)
- Mean: (30+35+40+45+1000) ÷ 5 = 230K (seems high!)
- Median: 40K (more realistic!)
The one millionaire didn’t mess up the median!
🎯 MODE
The “Most Popular” Number
What Is It?
The mode is the number that appears most often.
👟 Look at the shoe sizes in a store’s bestseller list: 8, 9, 9, 9, 10, 10, 11
Size 9 appears 3 times — more than any other! Mode = 9
Rules for Mode
- One mode (Unimodal): One number appears most
- Two modes (Bimodal): Two numbers tie for most
- No mode: All numbers appear equally
Example 1: Finding the Mode
Favorite colors votes: Red, Blue, Blue, Green, Blue, Red
| Color | Count |
|---|---|
| Red | 2 |
| Blue | 3 ← Most! |
| Green | 1 |
✅ Mode = Blue
Example 2: Bimodal Data
Test scores: 70, 80, 80, 90, 90, 100
- 80 appears 2 times
- 90 appears 2 times
✅ Modes = 80 and 90 (Bimodal)
Example 3: No Mode
Ages: 10, 20, 30, 40, 50
Each number appears exactly once. No repeats!
✅ No Mode
📊 MODAL CLASS
The “Most Popular Range” in Grouped Data
What Is It?
When data is grouped into ranges (like 0-10, 11-20, 21-30), you can’t find exact mode. Instead, find the modal class — the range with the most values.
📝 Exam scores grouped:
- 0-40: 5 students
- 41-60: 8 students
- 61-80: 15 students ← Most!
- 81-100: 12 students
Modal Class = 61-80
Example: Heights of Students
| Height Range (cm) | Frequency |
|---|---|
| 140-150 | 4 |
| 150-160 | 10 |
| 160-170 | 18 |
| 170-180 | 8 |
Which range has the highest frequency?
160-170 has 18 students — the most!
✅ Modal Class = 160-170 cm
Modal Class Formula (For the Math Nerds!)
If you need the exact mode within the modal class:
Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
Where:
- L = Lower boundary of modal class
- f₁ = Frequency of modal class
- f₀ = Frequency of class before
- f₂ = Frequency of class after
- h = Class width
Example using our height data:
- Modal class: 160-170
- L = 160, h = 10
- f₀ = 10, f₁ = 18, f₂ = 8
Mode = 160 + [(18-10) / (2×18 - 10 - 8)] × 10 Mode = 160 + [8 / 18] × 10 Mode = 160 + 4.44 Mode ≈ 164.44 cm
🎭 When to Use Each?
graph TD A[What do you want to know?] --> B{Type of Question} B --> C[Fair Share?] B --> D[Middle Value?] B --> E[Most Common?] C --> F[Use MEAN] D --> G[Use MEDIAN] E --> H[Use MODE] F --> I[Good for: Grades, Prices] G --> J[Good for: Salaries, Ages] H --> K[Good for: Sizes, Votes]
Quick Decision Guide
| Situation | Best Choice | Why? |
|---|---|---|
| Finding average test score | Mean | Fair representation |
| Finding typical salary | Median | Ignores extreme values |
| Finding popular shoe size | Mode | Most purchased |
| Grades with different weights | Weighted Mean | Accounts for importance |
| Merging two class averages | Combined Mean | Correct total average |
| Grouped data ranges | Modal Class | Most frequent range |
🌟 The Big Picture
Think of your three statistical friends:
- 🧮 Mean — The mathematician who adds everything and shares equally
- 📍 Median — The fair judge who finds the exact middle
- 🎯 Mode — The popularity counter who finds the favorite
Each one tells a different part of the story. A wise data detective uses all three to truly understand the numbers!
💡 Pro Tips
- Mean is sensitive to outliers (extreme values)
- Median is robust — it doesn’t budge for outliers
- Mode can be multiple or not exist at all
- For skewed data, median often tells the real story
- Weighted mean is essential when things have different importance
- Combined mean saves you from recalculating everything!
You’ve just learned the three wise averages! Now go summarize some data! 🎉