Average – Quantitative Aptitude (APO Exam)
Average (also called mean) is a measure of central tendency that summarises a set of numbers by a single value.
For the APO exam you’ll encounter the simple mean, weighted mean, harmonic mean (for rates), and “average speed” problems.
1. Simple Explanation
| Concept | What it is | When to use |
|---|---|---|
| Simple Mean | Sum of all observations ÷ number of observations | When every item contributes equally |
| Weighted Mean | Σ(value × weight) ÷ Σ(weight) | When items have different importance or frequency |
| Harmonic Mean | N ÷ Σ(1/values) | When dealing with rates (e.g., average speed, speed‑time problems) |
2. Key Points
| # | Point | Detail |
|---|---|---|
| 1 | Formula for Simple Mean | (\displaystyle \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}) |
| 2 | Formula for Weighted Mean | (\displaystyle \bar{x}w = \frac{\sum{i=1}^{n} x_i w_i}{\sum_{i=1}^{n} w_i}) |
| 3 | Formula for Harmonic Mean | (\displaystyle H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}) |
| 4 | Average Speed | If distance is same for each segment: (\displaystyle \bar{v} = \frac{n}{\sum_{i=1}^{n} \frac{1}{v_i}}) |
| 5 | Average Rate (when time differs) | (\displaystyle \bar{r} = \frac{\text{Total quantity}}{\text{Total time}}) |
| 6 | Common Pitfalls | • Mixing units (e.g., km and mi). • Forgetting to convert time to same unit. • Using simple mean for weighted data. |
| 7 | Typical APO Questions | • Average of given numbers. • Average of two speeds over equal distances. • Weighted average of grades or test scores. |
3. Worked Examples
Example 1 – Simple Average
Question: Find the average of 12, 15, 18, 20, and 25.
Solution
[
\bar{x} = \frac{12+15+18+20+25}{5} = \frac{90}{5} = 18
]
Answer: 18
Example 2 – Weighted Average
Question: A class has 20 students in Class A and 30 students in Class B.
The average scores are 78 for Class A and 85 for Class B.
Find the overall class average.
Solution
[
\bar{x}_w = \frac{(78 \times 20) + (85 \times 30)}{20+30}
= \frac{1560 + 2550}{50}
= \frac{4110}{50}
= 82.2
]
Answer: 82.2
Example 3 – Harmonic Mean (Average Speed)
Question: A cyclist travels 60 km at 15 km/h and then another 60 km at 20 km/h.
What is the average speed?
Solution
Distances are equal, so use harmonic mean of speeds:
[
H = \frac{2}{\frac{1}{15} + \frac{1}{20}}
= \frac{2}{\frac{4}{60} + \frac{3}{60}}
= \frac{2}{\frac{7}{60}}
= \frac{2 \times 60}{7}
= \frac{120}{7}
\approx 17.14 \text{ km/h}
]
Answer: ≈ 17.14 km/h
Example 4 – Average Rate (Different Times)
Question: A factory produced 300 units in 5 hours and 150 units in 3 hours.
What is the overall average production rate (units per hour)?
Solution
Total units = 300 + 150 = 450
Total time = 5 + 3 = 8 h
[
\bar{r} = \frac{450}{8} = 56.25 \text{ units/hour}
]
Answer: 56.25 units/hour
4. Quick Revision Summary
- Simple Mean:
sum ÷ count– use when all items equal. - Weighted Mean:
(value × weight) ÷ sum of weights– use when different importance/frequency. - Harmonic Mean:
count ÷ Σ(1/value)– ideal for rates (speed, density, etc.) especially with equal distances. - Average Speed (equal distances):
2 ÷ (1/v1 + 1/v2)for two speeds. - Average Rate (different times):
total quantity ÷ total time. - Check Units: make sure all units are consistent (e.g., km/h, minutes).
- Common Mistake: using simple mean for weighted data or vice versa.
Practice Problems (Try before checking)
- The average of five numbers is 30. If four of the numbers are 28, 32, 25, and 35, find the fifth number.
- 12 students scored 70, 8 students scored 80, and 5 students scored 90. Find the overall average score.
- A train