🎯 Comparing Averages: Mean, Median & Mode
The Three Best Friends Who See Things Differently
Imagine three friends—Mean, Median, and Mode—are asked to describe a group of 5 kids by their pocket money:
₹10, ₹10, ₹10, ₹20, ₹50
Each friend has their own way of finding the “typical” amount!
🧮 Mean, Median, Mode Relationship
Mean Says: “Add Everything, Share Equally!”
Mean adds up all the money and divides by 5:
(10 + 10 + 10 + 20 + 50) ÷ 5 = ₹20
Mean is like a fair sharing robot—everyone gets the same slice.
Median Says: “Stand in Line, I Pick the Middle!”
Median arranges everyone from smallest to largest and picks the middle person:
10, 10, 10, 20, 50 → ₹10
Median is like a referee—standing right in the center.
Mode Says: “Who Shows Up the Most?”
Mode looks for the most common value:
10 appears 3 times → ₹10
Mode is like a popularity counter—whoever appears most wins!
🔍 The Big Relationship Discovery
graph TD A[Data Shape] --> B{Is it Symmetric?} B -->|Yes| C[Mean ≈ Median ≈ Mode] B -->|No| D{Which way is the tail?} D -->|Right tail| E[Mean > Median > Mode] D -->|Left tail| F[Mean < Median < Mode]
Symmetric Data (Bell-shaped)
When data is perfectly balanced like a see-saw:
Mean = Median = Mode
Example: Test scores: 70, 75, 80, 85, 90
- Mean = 80, Median = 80, Mode = (none, but center is 80)
Right-Skewed Data (Tail stretches right)
When a few really big values pull the mean up:
Mean > Median > Mode
Example: Salaries at a small company: ₹30K, ₹35K, ₹35K, ₹40K, ₹200K (CEO)
- Mode = ₹35K (most common)
- Median = ₹35K (middle value)
- Mean = ₹68K (pulled up by CEO’s salary!)
Left-Skewed Data (Tail stretches left)
When a few really small values drag the mean down:
Mean < Median < Mode
Example: Age at retirement party (most are old, one young intern): 25, 58, 60, 60, 62
- Mean = 53, Median = 60, Mode = 60
🎯 Choosing the Right Central Tendency
Think of choosing an average like picking the right tool from a toolbox!
🔧 When to Use Mean
✅ Use Mean when:
- Data is symmetric (no extreme values)
- You need mathematical calculations later
- All values are equally important
Example: Your 5 test scores: 85, 88, 90, 87, 90
Mean = 88 → Great choice! No outliers pulling it around.
❌ Avoid Mean when:
- There are extreme outliers
- Data is heavily skewed
📏 When to Use Median
✅ Use Median when:
- Data has outliers or extreme values
- Data is skewed
- You want the “typical” middle person
Example: House prices in a neighborhood: ₹40L, ₹45L, ₹50L, ₹55L, ₹500L (mansion)
Median = ₹50L → Much better than Mean (₹138L)!
Real-world uses:
- Income reports
- House prices
- Waiting times at hospitals
🏆 When to Use Mode
✅ Use Mode when:
- Data is categorical (colors, brands, choices)
- You want the most popular option
- Finding the “bestseller”
Example: Favorite ice cream flavors: Vanilla (15), Chocolate (22), Strawberry (8)
Mode = Chocolate → The crowd favorite!
Real-world uses:
- Shoe sizes to stock
- Most ordered dish
- Popular baby names
📊 Quick Decision Flowchart
graph TD A[What type of data?] --> B{Categorical?} B -->|Yes| C[Use MODE 🏆] B -->|No| D{Has outliers?} D -->|Yes| E[Use MEDIAN 📏] D -->|No| F{Need calculations?} F -->|Yes| G[Use MEAN 🧮] F -->|No| H[Mean or Median both OK]
⚙️ Properties of Averages
Each average has superpowers and weaknesses!
Mean’s Properties
| Property | Explanation | Example |
|---|---|---|
| Uses all values | Every number counts | Change any value, mean changes |
| Affected by outliers | Extreme values pull it | ₹10K salaries + ₹1M boss = high mean |
| Sum property | Sum = Mean × Count | If mean is 5 for 10 items, sum = 50 |
| Zero-sum deviations | Distances balance out | (Values - Mean) always sum to 0 |
Example of Sum Property: If a class of 30 students has mean score 75:
Total marks = 30 × 75 = 2,250
Median’s Properties
| Property | Explanation | Example |
|---|---|---|
| Resistant to outliers | Extreme values don’t affect it much | Millionaire joins; median barely moves |
| Position-based | Only cares about middle | Changing extreme values doesn’t matter |
| Divides data in half | 50% above, 50% below | Half the class scored above median |
Example: Data: 2, 4, 6, 8, 100
- Median = 6 (unchanged even with outlier 100!)
- Mean = 24 (pulled up massively)
Mode’s Properties
| Property | Explanation | Example |
|---|---|---|
| Can be multiple | Two modes = bimodal | Scores: 70, 70, 85, 85 (two modes!) |
| Can be none | When all values appear once | All different values = no mode |
| Works with categories | Only average for text data | Favorite color: “Blue” is the mode |
| Not affected by extremes | Outliers don’t change popularity | Same bestseller regardless of extremes |
🎪 The Circus Trick: Comparing All Three
Imagine a circus with 5 performers and their heights (in cm):
150, 150, 160, 170, 220 (the giant!)
| Measure | Value | What it tells us |
|---|---|---|
| Mean | 170 cm | The “fair share” height |
| Median | 160 cm | The middle performer |
| Mode | 150 cm | Most common height |
Since Mean > Median > Mode, the data is right-skewed (the giant pulls the mean up!).
🌟 Golden Rules to Remember
-
Skew tells the story:
- Right skew → Mean is highest
- Left skew → Mean is lowest
- Symmetric → All three are friends (equal!)
-
Outliers hate Mean, love Median
- One billionaire in a village? Use Median for “typical” income!
-
Categories need Mode
- Can’t calculate mean of “Red, Blue, Green”!
-
Mean is a team player
- It considers everyone’s contribution equally.
🎯 Real-Life Superhero Scenario
The School Test Dilemma:
Your class scores: 45, 50, 55, 60, 95
- Mean = 61 → Teacher says “Average is good!”
- Median = 55 → “Half the class is below 55”
- Mode = None → All scores are different
Which average should the principal see?
Median (55) gives the honest picture. The topper’s 95 inflates the mean!
🚀 You’ve Got This!
Now you know:
- ✅ How Mean, Median, and Mode relate to each other
- ✅ When to pick each one like a pro
- ✅ What makes each average special (and problematic!)
The next time someone shows you data, you’ll know exactly which average tells the real story! 🎉