One-Line Definition
An Average (Arithmetic Mean) is the single value that represents the equal distribution of a sum across a group of observations.
Water Level Mental Model
Imagine 4 glasses with 10ml, 20ml, 30ml, and 40ml of water:
- • Pour all 4 into 1 large pitcher → Total = 100ml
- • Pour back equally into 4 glasses → Each glass gets 25ml
- • The average level of 25ml levels out all peaks and valleys.
Averages form the basis of Data Interpretation tables, Alligation mixtures, cricket batting averages, and replacement word problems across all MBA exams:
Weighted Average Formula
Combines two groups of sizes n1 and n2 with averages A1 and A2.
Assumed Mean Deviation Shortcut
Saves time by avoiding long addition of large numbers.
SHORTCUT RULES & KEY FORMULAS
Person Joins / Leaves / Replaced
When an old member is replaced by a new member, the total count N remains constant.
AP Series Average Shortcut
Applies to any sequence with constant common difference (consecutive integers, odd, even, multiples).
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: Find the average of 10, 15, 20, 25, and 30.
Step 1: Sum = 10 + 15 + 20 + 25 + 30 = 100.
Step 2: Count = 5 → Average = 100 ÷ 5 = 20.
Q: The average marks of 8 students are 72. A 9th student joins the group with 90 marks. Find the new average.
Step 1: Initial Sum = 8 × 72 = 576.
Step 2: New Total Sum = 576 + 90 = 666.
Step 3: New Average = 666 ÷ 9 = 74.
Q: The average age of 12 employees is 30 years. One employee aged 42 leaves and a new employee joins, causing the average to drop by 1 year. Find the age of the new employee.
Step 1: Total Count N = 12 (constant).
Step 2: Drop in total sum = N × Drop in Avg = 12 × 1 = 12 years.
Step 3: New Age = Old Age - Drop = 42 - 12 = 30 years.
Mistake 1
Direct Averaging of Averages
Taking (A1 + A2)/2 when group sizes n1 and n2 are unequal. You MUST use Weighted Average Aw = (n1·A1 + n2·A2)/(n1 + n2).
Mistake 2
Ignoring Count Adjustments
Forgetting to change the divisor N when people join (N + 1) or leave (N - 1). Check total count carefully!
Mistake 3
Confusing Sum with Average
Mixing up total sum of ages/scores with individual average when setting up algebraic equations.
1. Find the average of the first 20 positive odd integers.
Answer: Average of first n odd numbers = n = 20.
2. The average age of a family of 5 members is 24 years. If the age of the youngest member is 8 years, find the average age of the family at the time of his birth.
Answer: Current Sum = 120. Total sum 8 yrs ago for 4 members = 120 - (5×8) = 80 → Average = 80 / 4 = 20 years.
3. In a class of 60 students, the ratio of boys to girls is 3 : 2. The average score of boys is 70 and girls is 80. Find the overall class average.
Answer: Aw = (3×70 + 2×80) / (3+2) = (210 + 160) / 5 = 370 / 5 = 74.
FREQUENTLY ASKED QUESTIONS
❓ Can you average two average values directly?
No! You can only average two averages directly if both groups have the exact same number of observations. Otherwise, you must use the Weighted Average formula Aw = (n1·A1 + n2·A2) / (n1 + n2).
❓ How does the Deviation Method save time in CAT?
Instead of summing large multi-digit numbers, pick an assumed mean near the middle, calculate small positive and negative deviations, sum them, and divide by the total count.
❓ How do you find the average of consecutive numbers instantly?
For any arithmetic progression (consecutive integers, even numbers, or multiples), Average = (First Term + Last Term) / 2. If the count of terms is odd, it equals the exact middle term.