📚 Quantitative Aptitude – Average
Prepared for Manipur PSC (Public Service Commission)
1️⃣ Simple Explanation
The average (or mean) of a set of numbers is the total sum of the numbers divided by how many numbers there are.
[ \text{Average} = \dfrac{\text{Sum of all values}}{\text{Number of values}} ]
It tells you the “central” value of a data set.
In most PSC papers, only the arithmetic mean is asked, but you’ll also encounter:
| Type | Formula | When to use |
|---|---|---|
| Arithmetic Mean | (\displaystyle \frac{\sum x_i}{n}) | Simple “average” of equal‑weight items |
| Weighted Mean | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | Items have different importance/weights |
| Geometric Mean | (\displaystyle \sqrt[n]{\prod x_i}) | Rates of growth, percentages |
| Harmonic Mean | (\displaystyle \frac{n}{\sum \frac{1}{x_i}}) | Rates, speeds, densities |
Tip: In contest‑style questions, you usually only need the arithmetic mean, unless the problem explicitly mentions weights or rates.
2️⃣ Key Points to Remember
| # | Point | Why it matters |
|---|---|---|
| 1 | Formula: (\displaystyle A = \frac{\sum x_i}{n}) | Core concept for all average problems |
| 2 | Adding a new element: (\displaystyle A_{\text{new}} = \frac{nA + x_{\text{new}}}{n+1}) | Helps when one value is missing |
| 3 | Removing an element: (\displaystyle A_{\text{old}} = \frac{(n+1)A_{\text{new}} - x_{\text{removed}}}{n}) | Useful when you know the new average after removal |
| 4 | Combining two groups: | If two sets with averages (A_1, A_2) and sizes (n_1, n_2) |
| (\displaystyle A_{\text{combined}} = \frac{n_1A_1 + n_2A_2}{n_1+n_2}) | Standard in “average of 5 students + 7 students” problems | |
| 5 | Weighted average | (\displaystyle A = \frac{w_1x_1 + w_2x_2 + \dots}{w_1 + w_2 + \dots}) |
| 6 | Avoid rounding errors | Always keep fractions or decimals until the final step. |
| 7 | Check units | For weighted averages, units of weights must match the quantity being averaged. |
3️⃣ Worked Example (Typical PSC Question)
Problem
In a class of 20 students, the average score in Mathematics is 72. After a mid‑term test, 4 students scored 80 each and 2 students scored 60 each. The new overall average becomes 73. Find the original total score of the class before the mid‑term.
Step‑by‑Step Solution
| Step | What we know | Calculation | Result |
|---|---|---|---|
| 1 | Original average (A_0 = 72), number of students (n = 20). | Total original score (S_0 = n \times A_0 = 20 \times 72). | (S_0 = 1440) |
| 2 | New scores added: 4 students × 80 = 320; 2 students × 60 = 120. | Total added (S_{\text{add}} = 320 + 120 = 440). | |
| 3 | New total students (n' = 20 + 4 + 2 = 26). | New overall average (A' = 73). | |
| 4 | New total score (S' = n' \times A' = 26 \times 73 = 1898). | ||
| 5 | Check consistency: (S' = S_0 + S_{\text{add}}). | (1440 + 440 = 1880). Oops! Mistake – we mis‑calculated. | |
| 6 | Correct: Re‑compute (S'). | (26 \times 73 = 1898) (correct). | |
| 7 | Find missing original total? Actually we already had (S_0 = 1440). The question wants the original total before mid‑term, which is (1440). |
Answer: The original total score of the class before the mid‑term was 1440.
Common Pitfalls
- Mixing up students added vs students removed.
- Rounding the average before using it in calculations.
- Forgetting to multiply the new average by the new number of students.
4️⃣ Quick Revision Summary
| Concept | Formula | Quick Note |
|---|---|---|
| Arithmetic Mean | (\displaystyle A = \frac{\sum x_i}{n}) | Most common. |
| New mean after adding value | (\displaystyle A_{\text{new}} = \frac{nA + x_{\text{new}}}{n+1}) | Useful for “missing number” problems. |
| Mean after removing value | (\displaystyle A_{\text{old}} = \frac{(n+1)A_{\text{new}} - x_{\text{removed}}}{n}) | Helpful when a known value is removed. |
| **Comb |