📊 Average - Complete Theory

Master Average calculations - foundation for Data Interpretation!

🎯 What is Average?

Average is the central value of a set of numbers.

Formula:

Average = Sum of all values / Number of values

Average = Total / Count

Key Concept: Average represents the “mean” or “typical” value.

📐 Basic Formulas

Formula 1: Basic Average

If numbers are a₁, a₂, a₃, ..., aₙ

Average = (a₁ + a₂ + a₃ + ... + aₙ) / n

Example: Average of 10, 20, 30, 40, 50

Average = (10 + 20 + 30 + 40 + 50) / 5
        = 150 / 5
        = 30

Formula 2: Finding Sum from Average

Sum = Average × Number of values

If Average = A and Count = n
Sum = A × n

Example: Average of 5 numbers is 40. Find sum.

Sum = 40 × 5 = 200

Formula 3: Finding Count from Average

Number of values = Sum / Average

Count = Total / Average

💡 Important Concepts

1. Average of Consecutive Numbers

Natural Numbers (1, 2, 3, …n):

Average = (n + 1) / 2

Example: Average of 1 to 100
= (100 + 1) / 2 = 50.5

Any Consecutive Numbers (a to b):

Average = (First + Last) / 2
        = (a + b) / 2

Example: Average of 21 to 30
= (21 + 30) / 2 = 25.5

2. Average of Arithmetic Progression (AP)

For AP series:

Average = (First term + Last term) / 2

Example: 5, 10, 15, 20, 25
Average = (5 + 25) / 2 = 15

3. Average of Consecutive Even/Odd Numbers

Even Numbers (2, 4, 6, …2n):

Average = (First + Last) / 2 = n + 1

Example: 2, 4, 6, 8, 10
Average = (2 + 10) / 2 = 6

Odd Numbers (1, 3, 5, …, 2n-1):

Average = (First + Last) / 2 = n

Example: 1, 3, 5, 7, 9
Average = (1 + 9) / 2 = 5

🔄 Change in Average

When New Value is Added

New Average = Old Average + (Added Value - Old Average) / (n + 1)

Or simpler:
New Sum = Old Sum + Added Value
New Average = New Sum / New Count

Example: Average of 5 numbers is 20. If 30 is added, find new average.

Old Sum = 20 × 5 = 100
New Sum = 100 + 30 = 130
New Average = 130 / 6 = 21.67

When Value is Removed

New Average = (Old Sum - Removed Value) / (n - 1)

Example: Average of 6 numbers is 25. If one number 40 is removed, find new average.

Old Sum = 25 × 6 = 150
New Sum = 150 - 40 = 110
New Average = 110 / 5 = 22

When Value is Replaced

New Average = Old Average + (New Value - Old Value) / n

Example: Average of 5 numbers is 30. One number 25 is replaced by 40. Find new average.

Change = 40 - 25 = 15
New Average = 30 + (15 / 5) = 30 + 3 = 33

⚖️ Weighted Average

When different values have different “weights” (importance):

Weighted Average = (w₁ × v₁ + w₂ × v₂ + ... + wₙ × vₙ) / (w₁ + w₂ + ... + wₙ)

Where:
w = weight (frequency, importance)
v = value

Example: Student scored 80 in Quant (3 credits), 70 in English (2 credits), 90 in Reasoning (3 credits). Find average.

Weighted Average = (3×80 + 2×70 + 3×90) / (3 + 2 + 3)
                 = (240 + 140 + 270) / 8
                 = 650 / 8
                 = 81.25

💡 Solved Examples

Example 1: Basic Average

Q: Find average of 15, 25, 35, 45, 55.

Solution:

Sum = 15 + 25 + 35 + 45 + 55 = 175
Count = 5

Average = 175 / 5 = 35

Answer: 35

Example 2: Finding Missing Number

Q: Average of 5 numbers is 40. Four numbers are 35, 38, 42, 45. Find the fifth number.

Solution:

Sum of 5 numbers = 40 × 5 = 200
Sum of 4 known numbers = 35 + 38 + 42 + 45 = 160

Fifth number = 200 - 160 = 40

Answer: 40

Example 3: Age Problems

Q: Average age of 10 students is 15 years. If teacher’s age (45 years) is included, find new average.

Solution:

Total age of 10 students = 15 × 10 = 150
Total age with teacher = 150 + 45 = 195
Total people = 10 + 1 = 11

New Average = 195 / 11 = 17.73 years (approx)

Answer: 17.73 years

Example 4: Consecutive Numbers

Q: Find average of first 50 natural numbers.

Solution:

Method 1: Formula
Average = (n + 1) / 2 = (50 + 1) / 2 = 25.5

Method 2: First + Last
Average = (1 + 50) / 2 = 25.5

Answer: 25.5

Example 5: Replacement

Q: Average of 6 numbers is 30. One number 36 is replaced by 48. Find new average.

Solution:

Change = 48 - 36 = 12
Per number change = 12 / 6 = 2

New Average = 30 + 2 = 32

Answer: 32

Example 6: Wrong Calculation

Q: Average of 10 numbers was calculated as 25. Later it was found that one number 30 was misread as 50. Find correct average.

Solution:

Wrong Sum = 25 × 10 = 250
Difference = 50 - 30 = 20 (excess)
Correct Sum = 250 - 20 = 230

Correct Average = 230 / 10 = 23

Answer: 23

Example 7: Speed Average

Q: Car travels 40 km at 30 km/hr and next 40 km at 60 km/hr. Find average speed.

Solution:

⚠️ Average speed ≠ (30 + 60) / 2 ❌

Average Speed = Total Distance / Total Time

Time₁ = 40 / 30 = 4/3 hours
Time₂ = 40 / 60 = 2/3 hours
Total Time = 4/3 + 2/3 = 2 hours

Average Speed = 80 / 2 = 40 km/hr

Answer: 40 km/hr

Shortcut for Equal Distances:

Average Speed = (2 × s₁ × s₂) / (s₁ + s₂)
              = (2 × 30 × 60) / (30 + 60)
              = 3600 / 90
              = 40 km/hr

📊 Special Patterns

Pattern 1: Continuous Series

First n natural numbers: (n + 1) / 2
First n even numbers: n + 1
First n odd numbers: n
Multiples of k from k to kn: k(n + 1) / 2

Examples:

Average of 1, 2, 3, ..., 100 = 50.5
Average of 2, 4, 6, ..., 100 = 51
Average of 1, 3, 5, ..., 99 = 50
Average of 5, 10, 15, ..., 100 = 52.5

Pattern 2: Grouping Method

If average of 'a' numbers is A and
   average of 'b' numbers is B:

Combined Average = (a×A + b×B) / (a + b)

Example: Average of 6 numbers is 20, average of 4 numbers is 30. Find combined average.

Combined = (6×20 + 4×30) / (6 + 4)
         = (120 + 120) / 10
         = 24

Pattern 3: Equal Distribution

If sum increases by 'x' and count increases by 'n':
New average increases by x/n

⚡ Quick Shortcuts

Shortcut 1: Consecutive Numbers

Average = Middle number (if odd count)
Average = Mean of two middle numbers (if even count)

Example: 11, 12, 13, 14, 15
Average = 13 (middle number)

Example: 10, 11, 12, 13
Average = (11 + 12) / 2 = 11.5

Shortcut 2: Deviation Method

Assume any value as average (usually middle value)
Find deviations from assumed value
True Average = Assumed Average + (Sum of Deviations / Count)

Example: Find average of 47, 51, 53, 49, 55

Assume Average = 50

Deviations: (47-50), (51-50), (53-50), (49-50), (55-50)
          = -3, +1, +3, -1, +5
Sum of Deviations = 5

Average = 50 + (5/5) = 50 + 1 = 51

Shortcut 3: First n Numbers Pattern

Sum of first n natural numbers = n(n+1)/2
Average = (n+1)/2

Memorize:
1 to 10 → Avg = 5.5
1 to 100 → Avg = 50.5
1 to 1000 → Avg = 500.5

⚠️ Common Mistakes

❌ Mistake 1: Average Speed

Wrong: Average speed = (s₁ + s₂) / 2 ✗
Right: Average speed = Total Distance / Total Time ✓

❌ Mistake 2: Weighted Average

Wrong: Treating all values equally when they have different frequencies ✗
Right: Use weighted average formula ✓

❌ Mistake 3: Adding Averages

Wrong: Average of two groups = (Avg₁ + Avg₂) / 2 ✗
Right: Must consider count of each group ✓
Combined = (n₁×Avg₁ + n₂×Avg₂) / (n₁ + n₂)

❌ Mistake 4: Even/Odd Consecutive

For 2, 4, 6, 8, 10:
Wrong: Average = 5 + 1 = 6 (using odd formula) ✗
Right: Average = (2 + 10) / 2 = 6 ✓

📝 Practice Problems

Level 1:

1. Find average of 12, 18, 24, 30, 36
2. Average of 8 numbers is 15. Find their sum.
3. Find average of first 20 natural numbers.

Level 2:

4. Average of 5 numbers is 40. If one number 50 is added, find new average.
5. Average of 10 numbers is 25. One number 20 was misread as 30. Find correct average.
6. Average of 6 numbers is 30. If first 3 have average 25, find average of last 3.

Level 3:

7. Average age of 40 students is 15 years. If teacher’s age is included, average becomes 16. Find teacher’s age.
8. Car travels at 40 km/hr for 2 hours and 60 km/hr for 3 hours. Find average speed.
9. Average of 20 numbers is 50. Later 2 numbers 45 and 55 are removed. Find new average.

🔗 Related Topics

Prerequisites:

• Percentage - For percentage-based average problems
• Ratio & Proportion - For weighted average

Related:

• Alligation & Mixture - Uses weighted average concept
• Time & Work - Average work done
• Data Interpretation - Average calculations in DI

Practice:

• Average Questions
• Formula Sheet

Master Average - It’s crucial for Data Interpretation section! 📊