📊 Average – Quantitative Aptitude Study Notes
(For Jharkhand Police Constable Exam – Quantitative Aptitude)
1. Simple Explanation
Average (also called Mean) is a measure of central tendency that tells you the “typical” value of a set of numbers.
Formula
[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} ]
In everyday life, averages help us compare groups, judge performance, and simplify data.
2. Key Points
| Concept | What to Remember | Why It Matters |
|---|---|---|
| Arithmetic Mean | Sum of numbers ÷ Count | Most common average used in exams |
| Weighted Mean | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | Handles different importance (e.g., marks in different subjects) |
| Mode & Median | Mode = most frequent; Median = middle value when sorted | Handy for quick checks; not required for PGCD but useful for practice |
| Common Mistakes | - Forgeting to include all values - Mixing up numerator/denominator |
Leads to wrong answers, especially in time‑pressure |
| Shortcut | For two numbers, average = ((a+b)/2) | Quick mental check |
| Unit Consistency | All terms must be in the same unit | Prevents errors when mixing units (e.g., km & m) |
| Trick Questions | “Average of 0 and 100” → 50 | Test basic understanding |
3. Worked Example
Problem
A police constable’s monthly performance scores (out of 100) for the last 5 months are: 78, 85, 92, 88, 76.
Find his average monthly score.
Step‑by‑Step
- List the scores
[ 78,; 85,; 92,; 88,; 76 ] - Sum the scores
[ 78 + 85 + 92 + 88 + 76 = 419 ] - Count the numbers
[ n = 5 ] - Apply the formula
[ \text{Average} = \frac{419}{5} = 83.8 ] - Answer
The average monthly score is 83.8 (you may round to 84 if the question requires an integer).
Quick Check
If you want a mental check:
- Pair 78 + 92 = 170 → average 85
- Pair 85 + 88 = 173 → average 86.5
- Remaining 76 → overall average close to 84 → matches 83.8.
4. Quick Revision Summary
- Mean = ( \frac{\text{Sum of all values}}{\text{Count}} ).
- Weighted mean uses weights: ( \frac{\sum w_i x_i}{\sum w_i} ).
- Always keep units the same.
- Verify you have counted every value.
- For two numbers, average = ((a+b)/2).
- Use mental pairing for quick checks.
- Typical exam pattern: 5–10 numbers, sometimes weighted (e.g., exam marks).
Practice Problem (Try It Yourself)
The constable’s monthly attendance percentages over 6 months are: 95%, 92%, 88%, 90%, 93%, 94%.
What is his average attendance?
(Answer: ( \frac{95+92+88+90+93+94}{6} = 91.83% ) → about 91.8%)
Tip: Use a calculator for large sums, but for quick mental math, round numbers to simplify and adjust at the end. Happy studying!