Average – Quantitative Aptitude
Program: Odisha Patwari / Lekhpal
Topic: Average (Mean)
1. Simple Explanation
- Average (Mean) is the sum of a group of numbers divided by the count of numbers.
- It gives a representative value that summarizes the whole data set.
- In the Patwari context, averages help in land‑valuation, tax calculation, and land‑record summaries.
2. Key Points
| Concept | Formula | Notes |
|---|---|---|
| Simple Mean | (\displaystyle \bar{x} = \frac{S}{n}) where (S) = sum of all values, (n) = number of values |
Most common. |
| Weighted Mean | (\displaystyle \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}) where (w_i) = weight (e.g., area, price) |
Used when values have different importance. |
| Average of Averages | If groups have equal size: (\displaystyle \bar{x}_{all}= \frac{\sum \bar{x}_i}{k}) Else: use weighted mean with group sizes as weights. |
Avoids double‑counting. |
| Incremental Average | After adding a new value (x_{n+1}): (\displaystyle \bar{x}{new}= \frac{n\bar{x}{old}+x_{n+1}}{n+1}) |
Useful when data arrives in sequence. |
| Common Pitfalls | • Mixing up sum and count. • Forgetting to include all items. • Using the wrong weight in weighted mean. |
Double‑check every step. |
3. Worked Example
Problem (Patwari style)
A farmer owns 4 plots of land with the following cultivable areas (in acres): 12, 18, 15, and 21. The market rates per acre for each plot are ₹2,500, ₹2,800, ₹2,600, and ₹2,700 respectively. Find:
- The simple average area of the plots.
- The weighted average price per acre, weighted by area.
Solution
1. Simple Average Area
[ S_{\text{area}} = 12 + 18 + 15 + 21 = 66 \text{ acres} ] [ n_{\text{plots}} = 4 ] [ \bar{A} = \frac{S_{\text{area}}}{n_{\text{plots}}} = \frac{66}{4} = 16.5 \text{ acres} ]
Answer: Average area = 16.5 acres.
2. Weighted Average Price per Acre
Weights = areas, values = prices per acre.
[ \text{Weighted sum of price} = \sum w_i x_i = (12)(2500) + (18)(2800) + (15)(2600) + (21)(2700) ]
Compute each term:
- (12 \times 2500 = 30,000)
- (18 \times 2800 = 50,400)
- (15 \times 2600 = 39,000)
- (21 \times 2700 = 56,700)
Sum: [ S_{\text{price}} = 30,000 + 50,400 + 39,000 + 56,700 = 176,100 ]
Total area (weight sum): [ W_{\text{total}} = 12 + 18 + 15 + 21 = 66 ]
Weighted mean price per acre: [ \bar{P}{w} = \frac{S{\text{price}}}{W_{\text{total}}} = \frac{176,100}{66} = 2,668.18 \text{ ₹/acre (rounded to 2 decimals)} ]
Answer: Weighted average price ≈ ₹2,668.18 per acre.
Quick Check
| Plot | Area (ac) | Price (₹/ac) | Weighted Price (₹) |
|---|---|---|---|
| 1 | 12 | 2,500 | 30,000 |
| 2 | 18 | 2,800 | 50,400 |
| 3 | 15 | 2,600 | 39,000 |
| 4 | 21 | 2,700 | 56,700 |
| Total | 66 | – | 176,100 |
Divide total weighted price by total area → ₹2,668.18.
4. Quick Revision Summary
- Mean = Total sum ÷ Count.
- Weighted Mean = Σ(weight × value) ÷ Σ(weight).
- For equal‑size groups, average of averages = simple mean of group means.
- Incremental mean: (\displaystyle \bar{x}{new}= \frac{n\bar{x}{old}+x_{new}}{n+1}).
- Always verify that sum and count are correct and that the right weight is used.
Tip for Patwari Exams:
- When the question mentions “average value of land per acre” → use weighted mean with area as weight.
- For “average number of acres per plot” → use simple mean.