Averages

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Averages — Key Facts for NABE (Pakistan)

• Simple Average: Sum of values ÷ Number of values
• Weighted Average: Σ(wᵢ × vᵢ) / Σwᵢ
• Average Speed: Total Distance ÷ Total Time (NOT average of speeds)
• Properties: Adding ‘k’ to all values increases average by ‘k’
• ⚡ Exam tip: Average speed questions are frequently misinterpreted — use Total Distance ÷ Total Time

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Averages — NABE (Pakistan) Study Guide

Basic Concept

An average is a single value that represents a set of values. It gives us a general idea of the data without looking at every individual value.

Formula for Simple Average (Mean):

Average = (Sum of all values) / (Number of values)

Notation: If values are x₁, x₂, x₃, …, xₙ:

x̄ = (x₁ + x₂ + x₃ + ... + xₙ) / n

Example Calculations

Example 1: Find the average of 12, 15, 18, 22, 25

• Sum = 12 + 15 + 18 + 22 + 25 = 92
• Number of values = 5
• Average = 92 ÷ 5 = 18.4

Example 2: Average of first 10 natural numbers

• Sum = n(n+1)/2 = 10 × 11/2 = 55
• Average = 55 ÷ 10 = 5.5
• Note: Average of first n natural numbers = (n+1)/2

Key Properties of Average

1. Addition Property: If you add ‘k’ to every value, the average increases by ‘k’
2. Subtraction Property: If you subtract ‘k’ from every value, the average decreases by ‘k’
3. Multiplication Property: If you multiply every value by ‘k’, the average multiplies by ‘k’
4. Division Property: If you divide every value by ‘k’, the average divides by ‘k’

Weighted Average

When values have different importance (weights):

Formula:

Weighted Average = (w₁v₁ + w₂v₂ + ... + wₙvₙ) / (w₁ + w₂ + ... + wₙ)

Example: A student scores 70 in terminal (weight 1), 80 in mid-term (weight 2), and 90 in final (weight 3). Find weighted average.

• Total weight = 1 + 2 + 3 = 6
• Weighted Average = (1×70 + 2×80 + 3×90) / 6
• = (70 + 160 + 270) / 6 = 500/6 ≈ 83.33

Average Speed

Critical Formula — Do NOT simply average the speeds!

Average Speed = Total Distance Covered / Total Time Taken

Example: A car travels 100 km at 50 km/h and returns 100 km at 80 km/h

• Time going = 100/50 = 2 hours
• Time returning = 100/80 = 1.25 hours
• Total Distance = 200 km
• Total Time = 3.25 hours
• Average Speed = 200/3.25 ≈ 61.54 km/h (NOT 65 km/h!)

NABE Exam Pattern

Common question types:

1. Finding missing numbers when average is given
2. Combined average of different groups
3. Average speed problems
4. Weighted average scenarios
5. Effect of adding/removing values on average

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Averages — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Mathematical Foundation

The arithmetic mean (average) minimizes the sum of squared deviations from a central value. This is why the average is used as the measure of central tendency — any other value would result in a larger total squared deviation.

Deviation from Mean: For any data set, the sum of deviations from the mean equals zero.

Σ(xᵢ - x̄) = 0

2. Finding Missing Values

When one value is missing: If average of n values is A, and one value x is replaced by y:

• New average = A’
• Total change = n × (A’ - A)
• Therefore: y - x = n × (A’ - A)

Example: Average of 5 numbers is 24. One number 18 is replaced by a new number. If new average is 26, find the new number.

• Total increase = 5 × (26 - 24) = 10
• New number = 18 + 10 = 28

When average is known and we add a value:

• New average of (n+1) values = A’
• Sum of n values = nA
• New value = (n+1)A’ - nA

3. Combined Average Problems

Formula for Two Groups: If group 1 has n₁ values with average A₁, and group 2 has n₂ values with average A₂:

Combined Average = (n₁A₁ + n₂A₂) / (n₁ + n₂)

Example: Section A of 30 students has average 70 marks; Section B of 20 students has average 80 marks. Find combined average.

• Combined = (30×70 + 20×80) / (30+20)
• = (2100 + 1600) / 50
• = 3700/50 = 74 marks

4. Average Speed — Complete Analysis

Why simple average fails: Consider two speeds: 30 km/h and 60 km/h each for 1 hour

• Simple average = 45 km/h (WRONG for average speed)
• Total distance = 30 + 60 = 90 km
• Total time = 2 hours
• Average speed = 90/2 = 45 km/h (This worked because equal time!)

Consider equal distance case:

• Distance d at 30 km/h: time = d/30
• Distance d at 60 km/h: time = d/60
• Total distance = 2d
• Total time = d/30 + d/60 = 3d/60 = d/20
• Average speed = 2d ÷ (d/20) = 40 km/h

Key Insight: Average speed = Harmonic Mean of speeds when distances are equal.

5. Effect of Adding/Removing Data on Average

Adding one value:

• Old average = A, old count = n
• New average = A’
• (n×A + x) / (n+1) = A’
• Therefore: x = (n+1)A’ - nA

Removing one value:

• Same formula rearranged

6. Special Averages

Average of First n Natural Numbers:

• Sum = n(n+1)/2
• Average = (n+1)/2

Average of First n Odd Numbers (1, 3, 5, …, 2n-1):

• Sum = n²
• Average = n

Average of First n Even Numbers (2, 4, 6, …, 2n):

• Sum = n(n+1)
• Average = n+1

Average of Squares of First n Natural Numbers:

• Sum = n(n+1)(2n+1)/6
• Average = (n+1)(2n+1)/6

7. Common Mistakes and How to Avoid Them

1. Arithmetic Mean vs. Geometric Mean: Don’t confuse! AM ≥ GM
2. Weighted vs. Simple Average: Use weighted when groups have different sizes
3. Average of Percentages: Generally requires weighted average
4. Speed Assumptions: Don’t assume equal speeds mean equal times

8. Advanced Problem Types

Problem with Changing Values:

• “The average of 10 numbers is 15. When a number is added, average becomes 17. Find the added number.”
• Solution: 11×17 - 10×15 = 187 - 150 = 37

Replacement Problems:

• “Average age of 30 students is 15. When teacher’s age is included, average becomes 16. Find teacher’s age.”
• Solution: 31×16 - 30×15 = 496 - 450 = 46 years

Practice Questions for NABE

1. The average of 8 numbers is 20. If each number is multiplied by 3, what is the new average?
2. The average weight of 8 men is 70 kg. A man weighing 75 kg leaves and is replaced by a new man weighing 65 kg. Find the new average.
3. A person travels from A to B at 40 km/h and returns at 60 km/h. Find average speed for the round trip.
4. The average marks of 30 students is 55. Later it was found that one entry marked as 48 should have been 84. Find the correct average.
5. The average of 5 consecutive numbers is n. Find these numbers in terms of n.

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