Average

Average — Study Notes for SSC CHSL

Overview

At its core, average represents the "central value" of a set of numbers. Mastering average problems is crucial because the concept appears both in standalone questions and as a building block for other topics like ratio, time-speed-distance, and data interpretation.

SSC CHSL tests three main variants: (1) basic mean calculation and reverse problems, (2) weighted average where different quantities contribute unequally, and (3) age-based average problems involving relationships between past, present, and future ages. The difficulty ranges from direct one-step calculations to multi-layered word problems requiring logical setup. Students must develop speed in formula application and accuracy in handling fractions and decimals.

Key Concepts

• Definition of Average: Average = (Sum of all observations) ÷ (Number of observations). It is the value each item would have if the total were distributed equally.
• Reverse Calculation: If average and count are known, the total sum = Average × Number of observations. This reverse formula is the basis for most SSC problems.
• Effect of Adding/Removing Values: When a new observation is added, the new sum = old sum + new value, then recalculate average with the updated count. Removing an observation works similarly in reverse.
• Weighted Average: When different groups have different counts, the combined average ≠ simple average of group averages. Instead, combined average = (Sum of all values from all groups) ÷ (Total count across all groups).
• Replacement and Correction: If one observation is replaced or corrected, the change in sum = (new value – old value). Adjust the total sum accordingly and recalculate average.
• Age-Based Averages: These problems mix the concept of average with time progression. Remember: if n years pass, each person's age increases by n, so the total age increases by n × (number of people).
• Shortcut – Deviation Method: When numbers are close to a reference value A, compute deviations from A, average the deviations, then add back to A. Saves time with large numbers.
• Increasing/Decreasing Average: If a new observation is greater than the current average, the new average increases. If it's less, the new average decreases. The magnitude of change depends on the count.

Formulas / Key Facts

1. Basic Average Formula: Average = Total Sum ÷ Count Sum = Average × Count
2. Weighted Average (Two Groups): Combined Average = (n₁×A₁ + n₂×A₂) ÷ (n₁ + n₂) where n₁, n₂ are counts and A₁, A₂ are group averages.
3. Effect of Replacement: New Sum = Old Sum + New Value – Old Value New Average = New Sum ÷ Count
4. Average of First n Natural Numbers: Average = (n + 1) ÷ 2
5. Average of First n Even Numbers: Average = n + 1
6. Average of First n Odd Numbers: Average = n
7. Age Progression: If average age of n people is A now, after t years average = A + t. Before t years, average was A – t.
8. Change in Average: If average increases by x when one observation is added, then: New Observation = Old Average + x(n + 1), where n is the original count.

Worked Examples

Example 1 (Basic Average) The average of 5 numbers is 28. If one number 40 is replaced by 25, what is the new average?

Solution: Original sum = 28 × 5 = 140. Change in sum = 25 – 40 = –15. New sum = 140 – 15 = 125. New average = 125 ÷ 5 = 25.

Example 2 (Weighted Average) A class has 30 boys with average weight 50 kg and 20 girls with average weight 45 kg. Find the average weight of the whole class.

Solution: Total weight of boys = 30 × 50 = 1500 kg. Total weight of girls = 20 × 45 = 900 kg. Total weight = 1500 + 900 = 2400 kg. Total students = 30 + 20 = 50. Average weight = 2400 ÷ 50 = 48 kg.

Example 3 (Age-Based Average) The average age of a family of 5 members is 24 years. If the age of the youngest member is 6 years, what was the average age of the family at the birth of the youngest member?

Solution: Present total age = 24 × 5 = 120 years. At the birth of the youngest (6 years ago), the youngest contributed 0 and the other 4 were each 6 years younger. Total age 6 years ago = 120 – (6 × 5) = 120 – 30 = 90 years. At that time, only 4 members existed (youngest just born, age = 0, but we count the moment of birth, so 4 members had aged). Actually: 6 years ago, there were only 4 members. Total age of those 4 members then = 120 – 6 (youngest's current age) – 6×4 (reduction in age of 4 members) = 120 – 6 – 24 = 90 years. Average = 90 ÷ 4 = 22.5 years.

Example 4 (Reverse Average) The average of 8 numbers is 15. If the average of the first 5 numbers is 12, what is the average of the remaining 3 numbers?

Solution: Total sum of 8 numbers = 15 × 8 = 120. Sum of first 5 numbers = 12 × 5 = 60. Sum of remaining 3 numbers = 120 – 60 = 60. Average of remaining 3 = 60 ÷ 3 = 20.

Common Mistakes

• Mistake: Taking simple average of averages → When combining groups, students often do (A₁ + A₂) ÷ 2 instead of the weighted formula. Fix: Always use total sum ÷ total count when merging groups with different sizes.
• Mistake: Forgetting to adjust the count when adding/removing items → Students recalculate sum correctly but divide by the old count. Fix: If one number is added, new count = old count + 1; if removed, count decreases by 1.
• Mistake: Incorrect age progression logic → Students add years only to the current total age, forgetting that each person ages. Fix: If n people and t years pass, total age increases by n × t, not just t.
• Mistake: Misinterpreting "average increases by x" → Students think the new observation equals old average + x. Fix: When average increases by x after adding one number, new observation = old average + x(old count + 1).
• Mistake: Sign errors in replacement problems → Confusing the direction of change (new – old vs. old – new). Fix: Always write change in sum = new value – old value, then adjust total accordingly.

Quick Reference

• Average = Sum ÷ Count ; Sum = Average × Count — the two-way formula is your foundation.
• Weighted Average ≠ Simple Average — always compute total sum across all groups then divide by total count.
• Replacement: New Sum = Old Sum + (New – Old). Recalculate average with same count.
• Age Problems: After t years, average age = current average + t. Before t years, average = current average – t.
• First n naturals: Average = (n+1)/2. First n evens: n+1. First n odds: n.
• Speed Trick: For numbers close to a base value, use deviation method to avoid large multiplications.