📚 Study Notes – Average
Exam: SBI Clerk (Quantitative Aptitude)
Topic: Average
A brief, clear, and practical guide to help you master the concept of average and solve related questions quickly.
1. Simple Explanation
Average (also called mean) is a measure that represents the central value of a set of numbers.
Formula
[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} ]
If the total of the numbers is S and there are n numbers, then
[
\text{Average} = \frac{S}{n}
]
Averages are used to answer questions like “What is the average score of a class?” or “What is the average speed of a vehicle?”
2. Key Points
| Point | Explanation |
|---|---|
| 1. Types of Averages | • Arithmetic Mean – Most common. • Weighted Average – When items have different importance (weights). • Median & Mode – Not part of “average” in the exam context. |
| 2. Arithmetic Mean | [ |
\text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} ] | | 3. Weighted Average | When each item has a weight (w_i):
[ \text{Weighted Mean} = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} ] | | 4. Adding/Removing Items | • Adding a number increases the sum but also increases the count.
• Removing a number decreases both. | | 5. Quick Tricks | • If new items are equal to the current average, the overall average stays unchanged.
• If new items are greater (or less) than the average, the overall average increases (or decreases). | | 6. Common Question Types | • “If one more student scores X, what is the new average?”
• “If the average of first k numbers is A, what is the average of first (k+1) numbers?”
• “Find the value of the missing number to make the average Y.” | | 7. Edge Cases | • When average is a fraction, you may need to use algebra or cross‑multiplication. | | 8. Rounding | SBI questions usually expect exact values or simple fractions; avoid unnecessary rounding unless asked. |
3. Worked Example
Problem
In a class of 8 students, the average score is 65. Two more students join the class and score 70 each. What is the new average?
Step‑by‑step Solution
-
Find the total score of the original 8 students.
[ \text{Sum}{\text{old}} = \text{Average}{\text{old}} \times \text{Count}_{\text{old}} = 65 \times 8 = 520 ] -
Add the scores of the two new students.
[ \text{Extra Sum} = 70 + 70 = 140 ] -
Compute the new total score.
[ \text{Sum}_{\text{new}} = 520 + 140 = 660 ] -
Compute the new number of students.
[ \text{Count}_{\text{new}} = 8 + 2 = 10 ] -
Calculate the new average.
[ \text{Average}_{\text{new}} = \frac{660}{10} = 66 ]
Answer: The new average score is 66.
4. Practice Question (Self‑Check)
The average of 15 numbers is 18. If three of these numbers are removed, and the average of the remaining 12 numbers is 20, what are the sum of the three removed numbers?
Solution Outline
- Total sum of all 15 numbers: (18 \times 15 = 270).
- Total sum of remaining 12 numbers: (20 \times 12 = 240).
- Sum of removed numbers: (270 - 240 = 30).
Answer: 30
5. Quick Revision Summary
| Topic | Formula / Key Idea |
|---|---|
| Arithmetic Mean | (\displaystyle \frac{\text{Sum of all numbers}}{\text{Count}}) |
| Weighted Mean | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) |
| Adding New Values | New average = (\frac{(old_sum + new_sum)}{(old_count + new_count)}) |
| Removing Values | New average = (\frac{(old_sum - removed_sum)}{(old_count - removed_count)}) |
| Trick | New items equal to current average → overall average unchanged. |
| Common Error | Mixing up sum and count; forgetting to update both. |
Final Tips
- Always write down both the sum and the count before manipulating averages.
- For weighted averages, keep track of the total weight separately.
- Use cross‑multiplication when the answer is a fraction.
- Practice similar questions under timed conditions to build speed.
Good luck with your SBI Clerk preparation! 🚀