State TET – Quantitative Aptitude
Topic: Average
Average (sometimes called mean) is the central value of a set of numbers. It is a very common topic in the State TET and appears in many sections (Arithmetic, Data Interpretation, and General Knowledge).
1. Simple Explanation
| Concept | Meaning | Symbol |
|---|---|---|
| Arithmetic Mean | Sum of all values ÷ Number of values | (\bar{x}) |
| Weighted Mean | Sum of (value × weight) ÷ Sum of weights | (\bar{x}_w) |
| Running / Moving Average | Mean of a fixed number of recent terms (often used in time‑series) | – |
- Average is a way to summarise a set of numbers with a single representative number.
- It gives an idea of the central tendency of the data.
- Misconceptions: “average” ≠ “median” or “mode”; it is a distinct measure.
2. Key Points
| # | Point | Details |
|---|---|---|
| 1 | Formula for Arithmetic Mean | (\displaystyle \bar{x} = \frac{S}{n}) where (S) = sum of all values, (n) = number of values. |
| 2 | Properties | • If you add a constant (c) to every term, the new mean = old mean + (c). • If you multiply every term by a constant (k), new mean = (k) × old mean. |
| 3 | Weighted Mean | (\displaystyle \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}) where (x_i) = value, (w_i) = weight. |
| 4 | Average Speed | (\displaystyle \text{Average Speed} = \dfrac{\text{Total Distance}}{\text{Total Time}}). |
| 5 | Average Rate of Change | (\displaystyle \frac{\Delta \text{Output}}{\Delta \text{Input}}). |
| 6 | Using Algebraic Manipulation | Often the problem gives partial sums or averages; use algebra to find missing values. |
| 7 | Common Traps | • Confusing average with median. • Using wrong denominator (total number of terms). • For weighted averages, forgetting to sum the weights. |
| 8 | Tip for TET | Problems may ask: “If the average of 10 numbers is 12, what is the average of the first 6 numbers if the remaining 4 are 15 each?” Use algebraic relationships. |
3. Worked Example 1: Simple Arithmetic Mean
Problem
The average of 8 numbers is 14. If 3 of those numbers are 20, 22, and 18, what is the average of the remaining 5 numbers?
Solution
- Let the sum of all 8 numbers be (S).
[ \bar{x} = \frac{S}{8} = 14 ;\Rightarrow; S = 14 \times 8 = 112 ] - Sum of the known numbers: (20 + 22 + 18 = 60).
- Sum of remaining 5 numbers: (S_{\text{rem}} = S - 60 = 112 - 60 = 52).
- Average of remaining 5 numbers: [ \bar{x}_{\text{rem}} = \frac{52}{5} = 10.4 ]
Answer: 10.4
4. Worked Example 2: Weighted Mean
Problem
A teacher grades 5 projects. The weights (importance) are 2, 3, 1, 4, and 2 respectively. The marks obtained are 75, 80, 65, 90, and 70. Find the weighted average score.
Solution
- Compute product of each mark and its weight:
| Mark ((x_i)) | Weight ((w_i)) | (w_i x_i) |
|---|---|---|
| 75 | 2 | 150 |
| 80 | 3 | 240 |
| 65 | 1 | 65 |
| 90 | 4 | 360 |
| 70 | 2 | 140 |
-
Sum of weighted values: (150 + 240 + 65 + 360 + 140 = 955).
Sum of weights: (2 + 3 + 1 + 4 + 2 = 12). -
Weighted mean: [ \bar{x}_w = \frac{955}{12} \approx 79.58 ]
Answer: Approximately 79.6 (to one decimal place).
5. Worked Example 3: Average Speed
Problem
A train travels 120 km to a station, then 80 km back to the starting point. It takes 3 hours to reach the station and 2 hours to return. What is the average speed for the whole journey?
Solution
- Total distance: (120 + 80 = 200) km.
- Total time: (3 + 2 = 5) hours.
- Average speed: [ \text{Avg Speed} = \frac{200}{5} = 40 \text{ km/h} ]
Answer: 40 km/h