📚 Quantitative Aptitude – Average
Subject: Himachal Pradesh Teacher Eligibility Test (TET)
Target: Quick, clear, and exam‑ready study notes
Average (Mean)
The average of a set of numbers is the sum of the numbers divided by the count of numbers. It represents a “central” value.
1️⃣ Simple Explanation
- Mean = (\displaystyle \frac{\text{Sum of all values}}{\text{Number of values}})
- Often called “arithmetic mean.”
- Gives a single number that “averages” the whole dataset.
2️⃣ Key Points & Formulas
| Concept | Formula | Notes |
|---|---|---|
| Mean of a set | (\displaystyle \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}) | (n) = number of items |
| Mean of two numbers | (\displaystyle \frac{a+b}{2}) | Quick when only two values |
| Changing the mean | If adding (k) to each of (n) numbers, new mean = old mean + (k) | No need to recompute sum |
| Weighted mean | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | Used when numbers have different importance |
| Average of averages | (\displaystyle \frac{n_1\bar{x}_1 + n_2\bar{x}_2 + \dots}{n_1+n_2+\dots}) | Weighted by counts |
3️⃣ Quick Tricks for TET
| Trick | When to use it | Example |
|---|---|---|
| Adding a constant to every element | If you add (k) to each of (n) numbers, mean increases by (k). | If mean of 5 numbers is 12, after adding 3 to each, new mean = 15. |
| Removing a number | New mean = (\displaystyle \frac{n\bar{x} - x_{\text{removed}}}{n-1}). | Mean = 10 for 4 numbers. Remove 8 → new mean = (\frac{4×10-8}{3}=9.\overline{3}). |
| Adding a number | New mean = (\displaystyle \frac{n\bar{x} + x_{\text{new}}}{n+1}). | Mean = 20 for 5 numbers. Add 30 → new mean = (\frac{5×20+30}{6}=20). |
| Use “average of averages” | When two or more groups are combined. | 3 students: 10, 12, 14 → mean = 12. 2 students: 18, 20 → mean = 19. Combined mean = (\frac{3×12 + 2×19}{5} = 15.4). |
| Sum from average | Sum = mean × count. | Mean = 8 for 10 numbers → sum = 80. |
4️⃣ Worked Example
Question (TET‑style):
A teacher recorded the marks of 6 students in a test: 78, 85, 92, 88, 75, 80.
After the test, the teacher added a bonus of 5 marks to every student.
What is the new average score?
Step‑by‑Step
-
Compute the original mean.
[ \text{Sum} = 78+85+92+88+75+80 = 498 ] [ \bar{x}_{\text{original}} = \frac{498}{6} = 83 ] -
Apply the bonus rule.
Adding 5 to every score increases the mean by 5.
[ \bar{x}_{\text{new}} = 83 + 5 = 88 ] -
Answer: The new average score is 88.
Why this is faster: No need to recompute the sum with bonuses.
5️⃣ Common Mistakes to Avoid
| Mistake | Fix |
|---|---|
| Adding the bonus to the mean instead of to the sum first. | Use the “add constant to each number” trick. |
| Forgetting to divide by the correct count after removing a number. | Re‑check denominator: (n-1). |
| Mixing up weighted mean when groups have different sizes. | Weight by the number of elements. |
| Assuming mean of averages equals average of all numbers. | Only true if groups have the same size. |
6️⃣ Quick Revision Summary
| Item | Detail |
|---|---|
| Definition | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) |
| Add constant | New mean = old mean + constant |
| Remove element | (\displaystyle \frac{n\bar{x} - x_{\text{rem}}}{n-1}) |
| Add element | (\displaystyle \frac{n\bar{x} + x_{\text{add}}}{n+1}) |
| Weighted average | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) |
| Average of averages | Weighted by group sizes |
| Tip | Whenever a number is added/removed, update the sum via formulas above rather than recomputing from scratch. |
📌 Final Tip
In TET, questions often involve quick updates (adding a constant, removing a student, etc.). Master the “update the mean” formulas – they save time and reduce errors.
Good luck with your preparation! 🚀