Average

Average

Overview

Average (or arithmetic mean) is one of the most reliable scoring areas in SBI Clerk Prelims. Questions are calculation-based with minimal tricks, making them ideal for securing quick marks if your fundamentals are solid.

To excel, you need three skills: applying the basic formula quickly, handling addition/removal of elements, and managing weighted averages. Most errors come from careless arithmetic or misreading "new average" vs "change in average" — both fixable with practice. Master this topic, and you'll find it pays dividends across the Numerical Ability section.

Key Concepts

• Basic Definition: Average = (Sum of all observations) ÷ (Number of observations). Rearrange freely: Sum = Average × Count.
• Adding a New Element: When a new value is added, the new average shifts toward that value. If the new value equals the old average, the average stays unchanged.
• Removing an Element: New Sum = Old Sum – Removed Value. Then divide by (n – 1) to get the new average.
• Replacement Effect: When one value is replaced by another, Change in Sum = (New Value – Old Value). This directly changes the total without altering the count.
• Weighted Average: When groups have different sizes, use (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂). Never simply average two averages unless group sizes are equal.
• Average of Consecutive Numbers: For any AP (arithmetic progression), Average = (First term + Last term) ÷ 2. For first n natural numbers, Average = (n + 1) ÷ 2.
• Age-Based Averages: When "x years ago" or "x years hence" is mentioned, each person's age changes by x. Total change = x × (number of people).

Formulas / Key Facts

Worked Examples

Example 1: Basic Average Calculation

Problem: The average of 5 numbers is 42. If one number 30 is replaced by 45, find the new average.

Solution:

• Original Sum = 42 × 5 = 210
• Change in Sum = 45 – 30 = +15
• New Sum = 210 + 15 = 225
• New Average = 225 ÷ 5 = 45

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Example 2: Adding a Person to a Group

Problem: The average age of 8 students is 15 years. A new student joins, and the average becomes 16 years. Find the age of the new student.

Solution:

• Original Sum = 8 × 15 = 120 years
• New Sum = 9 × 16 = 144 years
• Age of new student = 144 – 120 = 24 years

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Example 3: Weighted Average

Problem: In a class, 20 boys scored an average of 65 marks and 30 girls scored an average of 75 marks. Find the class average.

Solution:

• Total marks of boys = 20 × 65 = 1300
• Total marks of girls = 30 × 75 = 2250
• Combined Sum = 1300 + 2250 = 3550
• Total students = 20 + 30 = 50
• Class Average = 3550 ÷ 50 = 71 marks

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Example 4: Average Change Over Time

Problem: The average age of a family of 4 members is 25 years today. What will be the average age after 5 years?

Solution:

• After 5 years, each member is 5 years older.
• Total increase = 5 × 4 = 20 years
• Original Sum = 25 × 4 = 100 years
• New Sum = 100 + 20 = 120 years
• New Average = 120 ÷ 4 = 30 years

Shortcut: When time passes, average simply increases by that many years (if no member is added/removed). So, 25 + 5 = 30.

Common Mistakes

• Wrong Thinking: Averaging two group averages directly (e.g., boys avg 60, girls avg 80, so class avg is 70). Fix: Always use weighted average. Simple average works only when both groups have equal sizes.
• Wrong Thinking: Forgetting to change the count when a person joins or leaves. Fix: Adding someone → divide by (n + 1). Removing someone → divide by (n – 1). Write it explicitly.
• Wrong Thinking: Confusing "change in average" with "new average." Fix: If the question asks for the new average, calculate it fully. If it asks by how much the average changed, find the difference.
• Wrong Thinking: In replacement problems, subtracting the old value and adding the new value separately, then making arithmetic errors. Fix: Directly compute (New – Old) as a single step. This is the net change in sum.
• Wrong Thinking: In age problems, adding years to the average instead of to each individual. Fix: When time passes, add x years to EACH person. Total sum increases by x × n, so average increases by x. (This shortcut works, but understand why.)

Quick Reference

• Average = Sum ÷ Count — rearrange as needed.
• Replacement: Only sum changes, not count. New Sum = Old Sum + (New value – Old value).
• Weighted Average: (n₁A₁ + n₂A₂) ÷ (n₁ + n₂) — never average the averages blindly.
• First n natural numbers: Sum = n(n+1)/2, Average = (n+1)/2.
• Time shift in ages: Average increases by the same number of years as time passed (if group unchanged).
• Always verify: Does your answer make sense? New average should lie between the extremes.