In-Depth: Averages
CLAT Application & Relevance
Importance: VERY HIGH. Averages (specifically the arithmetic mean) are one of the three foundational pillars of Quantitative Techniques, alongside Percentages and Ratios. They are indispensable for analyzing data sets in CLAT, allowing you to find a representative value for a group of numbers, track performance, or infer missing information.
How it's tested: Calculating average profits/revenues from tables or line graphs; finding the average score/age of a group after adding or removing members (caselets); calculating weighted averages when different groups have different sizes.
Section 1: Core Concepts & Formulas
The average (or arithmetic mean) of a set of numbers is the sum of all values divided by the count of those values. It provides a single number that summarizes the data set.
Fundamental Formulas
- Simple Average (Mean) =
(Sum of all Observations) / (Number of Observations) - Crucial Rearrangement:
Sum of Observations = Average × Number of Observations. This is extremely useful for problems involving adding or removing elements from a group. - Median: The middle value in a sorted list of numbers. If there's an even count, it's the average of the two middle numbers. (Less common in CLAT QT, but good to know for general data literacy).
- Mode: The value that appears most frequently in a data set. (Very rare in CLAT QT).
Advanced Average Concepts
- Weighted Average: Used when different items or groups contribute differently to the overall average. Each item's value is multiplied by its weight (e.g., number of students in a section, quantity of an item).
Formula:Weighted Average = (Sum of (Weight × Value)) / (Sum of Weights)
Example:(w1*x1 + w2*x2 + ...) / (w1 + w2 + ...) - Average Speed: Total Distance / Total Time. (This is a specific application of average).
Section 2: Solved CLAT-Style Examples
Example 1: Average with Addition/Removal of Elements (Caselet)
Passage Context: The average age of 6 members of a legal team is 45 years. A new junior associate joins the team, and the average age of the team now becomes 43 years.
Question: "What is the age of the new junior associate?"
Detailed Solution:
1. Calculate Initial Sum of Ages:
Initial Sum = Average × Number = 45 years/member × 6 members = 270 years.
2. Calculate New Sum of Ages:
After the new associate joins, Number = 7 members. New Average = 43 years.
New Sum = New Average × New Number = 43 years/member × 7 members = 301 years.
3. Find the Age of the New Associate:
Age of New Associate = New Sum - Initial Sum = 301 - 270 = 31 years.
Answer: The age of the new junior associate is 31 years.
Example 2: Weighted Average (Table/Caselet)
Passage Context: A mock test was conducted for two batches of CLAT aspirants. Batch A has 80 students and their average score was 75. Batch B has 120 students and their average score was 60.
Question: "What is the overall average score of all students from both batches combined?"
Detailed Solution:
1. Calculate Sum of Scores for Batch A:
Sum_A = Average_A × Number_A = 75 × 80 = 6000.
2. Calculate Sum of Scores for Batch B:
Sum_B = Average_B × Number_B = 60 × 120 = 7200.
3. Calculate Total Sum of Scores:
Total Sum = Sum_A + Sum_B = 6000 + 7200 = 13200.
4. Calculate Total Number of Students:
Total Number = Number_A + Number_B = 80 + 120 = 200.
5. Calculate Overall Average:
Overall Average = Total Sum / Total Number = 13200 / 200 = 66.
Answer: The overall average score of all students from both batches is 66.