Andhra Pradesh Police Sub‑Inspector (SI) – Quantitative Aptitude
Topic: Average
1. Simple Explanation
The average (often called the arithmetic mean) of a set of numbers is the single value that represents the central tendency of that set.
Formula
[
\text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}}
]
It is the most common type of average used in exams, investigations, and everyday calculations.
2. Key Points
| Concept | What it Means | Formula | When to Use | Common Pitfalls |
|---|---|---|---|---|
| Arithmetic Mean | Sum of numbers ÷ count | ( \bar{x} = \frac{\sum x_i}{n} ) | General averages (marks, speeds, ages) | Forgetting to divide by n or mis‑counting items |
| Weighted Mean | Numbers have different importance (weights) | ( \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} ) | Exam marks with different marks, salary components | Mixing up weights and values; forgetting to sum weights |
| Harmonic Mean | Average of reciprocals (useful for rates) | ( H = \frac{n}{\sum (1/x_i)} ) | Average speed when distance is equal, rates of flow | Using it for normal data; forgetting to take reciprocal |
| Geometric Mean | Product of numbers raised to 1/n (for growth rates) | ( G = (\prod x_i)^{1/n} ) | Compound interest, population growth | Applies only to positive numbers; often mis‑used for general data |
| Average of Two Numbers | Quick shortcut | ( \frac{a+b}{2} ) | Quick checks | Forgetting to divide by 2 |
| Average Speed | Distance ÷ time (if distance is same for each segment) | ( v_{\text{avg}} = \frac{n\cdot d}{\sum t_i} ) | Travel, traffic analysis | Using simple mean of speeds instead of harmonic mean |
3. Worked Examples
Example 1: Arithmetic Mean of Five Numbers
Problem: Find the average of 12, 18, 25, 30, and 45.
Solution
- Sum: (12+18+25+30+45 = 130)
- Count: (n = 5)
- Average: (\displaystyle \frac{130}{5} = 26)
Answer: 26
Example 2: Weighted Average
Problem: A candidate’s marks are 70 (30% weight), 85 (40% weight), and 90 (30% weight). What is the weighted average?
Solution
- Multiply each mark by its weight:
- (70 \times 0.30 = 21)
- (85 \times 0.40 = 34)
- (90 \times 0.30 = 27)
- Sum of weighted marks: (21+34+27 = 82)
- Sum of weights: (0.30+0.40+0.30 = 1)
- Weighted mean: (\displaystyle \frac{82}{1} = 82)
Answer: 82
Example 3: Average Speed (Harmonic Mean)
Problem: A vehicle travels 120 km at 60 km/h, then 120 km at 80 km/h. Find the average speed for the whole journey.
Solution
- Total distance: (120+120 = 240) km
- Time for each segment:
- (t_1 = \frac{120}{60} = 2) h
- (t_2 = \frac{120}{80} = 1.5) h
- Total time: (2 + 1.5 = 3.5) h
- Average speed: (\displaystyle \frac{240}{3.5} = 68.57) km/h (≈ 69 km/h)
Answer: ~69 km/h
(Notice the use of harmonic mean because the distance is the same for each leg.)
4. Quick Revision Summary
| Type | Formula | Usage | Notes |
|---|---|---|---|
| Arithmetic Mean | ( \bar{x} = \dfrac{\sum x_i}{n} ) | General averages | Always divide by n |
| Weighted Mean | ( \bar{x}_w = \dfrac{\sum w_i x_i}{\sum w_i} ) | Marks, salary, components | Sum of weights must equal 1 or you must divide by ( \sum w_i ) |
| Harmonic Mean | ( H = \dfrac{n}{\sum |