Gujarat TET – Quantitative Aptitude
Topic: Average
Average (also called mean) is a measure of the central tendency of a set of numbers. It is the sum of all values divided by the number of values.
1️⃣ Simple Explanation
- Average = (Sum of all observations) ÷ (Number of observations)
- It gives a single representative value that summarizes a data set.
- Commonly used in everyday life: class marks, test scores, temperature, etc.
Example
Marks obtained by a student in 5 subjects: 78, 85, 92, 74, 88
Sum = 78 + 85 + 92 + 74 + 88 = 417
Count = 5
Average = 417 ÷ 5 = 83.4
2️⃣ Key Points to Remember
| Concept | What to Note |
|---|---|
| Notation | Average of n numbers x₁, x₂, …, xₙ is written as x̄ (x‑bar). |
| Formula | x̄ = (x₁ + x₂ + … + xₙ) / n |
| Units | Same as the original numbers (e.g., marks, meters, kilograms). |
| Effect of Outliers | A very high or very low value can skew the average. |
| Integer vs. Decimal | Result can be an integer or a decimal; always keep the decimal if it appears. |
| Comparing Averages | To compare two averages, look at the difference or percentage difference. |
| Average of Averages | If two groups have averages A₁ and A₂ with sizes n₁ and n₂, the combined average is: |
A = (A₁·n₁ + A₂·n₂) / (n₁ + n₂) |
|
| Weighted Average | When items have different importance (weights wᵢ): |
Avg = (Σ wᵢ·xᵢ) / Σ wᵢ |
|
| Common Pitfalls | 1) Adding the averages instead of weighted sums. 2) Forgetting to divide by the correct n. 3) Mixing units. |
| Applications in TET | Calculating class average, average marks of a student, average test scores, etc. |
3️⃣ Worked Example (Typical TET Question)
Question
A teacher has the following marks for 4 students in a class test: 78, 84, 72, and 90.
(a) Find the average marks of the class.
(b) If one more student joins the class with a score of 88, what will be the new average?
(a) Initial Average
Sum = 78 + 84 + 72 + 90 = 324
Count (n) = 4
Average = 324 / 4 = 81
Answer (a): The class average is 81 marks.
(b) New Average after adding 88
New sum = 324 + 88 = 412
New count = 4 + 1 = 5
New average = 412 / 5 = 82.4
Answer (b): The new average is 82.4 marks.
Key Observation
Adding a score higher than the current average (81) increases the average.
4️⃣ Quick Revision Summary
| Step | Action | Formula | Example |
|---|---|---|---|
| 1 | Sum all numbers | Σxᵢ | 78 + 85 + 92 + 74 + 88 = 417 |
| 2 | Count numbers | n | 5 |
| 3 | Divide sum by count | Average = Σxᵢ / n | 417 ÷ 5 = 83.4 |
| 4 | For combined groups | (A₁·n₁ + A₂·n₂) / (n₁ + n₂) | (80·4 + 90·2)/(4+2)=86.67 |
| 5 | For weighted items | (Σwᵢ·xᵢ) / Σwᵢ | (2·70 + 1·90)/3 = 76.67 |
Tips for TET:
- Always double‑check the number of terms (
n). - Keep units consistent (marks, percentages, etc.).
- For multiple-choice questions, compare averages by subtracting or using difference.
- Practice quick mental math: add numbers, then divide; remember common fractions (½, ⅓, ⅔, ¾).
Final Thought
Mastering averages equips you to solve a wide range of quantitative problems efficiently—an essential skill for the Gujarat TET exam. Good luck!