Average – Quantitative Aptitude (RRB ALP)
Average (also called arithmetic mean) is a fundamental concept used in many RRB ALP quantitative problems.
An average gives the “typical” value of a set of numbers.
1️⃣ Simple Explanation
- Average = Total of all values ÷ Number of values
- It is not influenced by the order of numbers, but it does change if any number is added or removed.
- In RRB questions you often need to:
- Find an average from a given sum or total.
- Compute a new average after adding or removing some numbers.
- Combine averages of two groups (weighted average).
- Work with averages of averages.
2️⃣ Key Points
| Concept | Formula / Idea | Example Use‑Case |
|---|---|---|
| Arithmetic Mean | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) | Average marks of 30 students. |
| New average after adding | (\displaystyle \bar{x}_{new} = \frac{n\bar{x} + m y}{n+m}) | Adding 5 new scores of 90 to a group of 10 scores. |
| New average after removing | (\displaystyle \bar{x}_{new} = \frac{n\bar{x} - k z}{n-k}) | Removing 3 scores of 70 from a group of 12 scores. |
| Weighted average of two groups | (\displaystyle \bar{x}_{total} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}) | Average salary of 5 employees vs 3 employees. |
| Average of averages (same number of items) | (\displaystyle \bar{x}_{total} = \frac{\bar{x}_1 + \bar{x}_2}{2}) | Two groups of 10 students each. |
| Average of averages (different sizes) | (\displaystyle \bar{x}_{total} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}) | 8 students with avg 75 vs 12 students with avg 80. |
| Common Pitfall | Forgetting to multiply the old average by the count before adding/removing numbers. | See worked example below. |
Quick Tricks
- Avoid recomputing the whole sum – use the old average and count.
- For “average of averages” – always weight by the number of elements, not by the average itself.
- When a number is removed, subtract it from the total sum before dividing by the new count.
3️⃣ Worked Example (Typical RRB Question)
Problem:
The average score of 20 students in a test is 68.
Three students who scored 60 each were excluded from the calculation.
What is the new average of the remaining 17 students?
Step‑by‑Step Solution
-
Find the total sum of the original 20 scores.
[ \text{Total}{20} = \bar{x}{20} \times 20 = 68 \times 20 = 1360 ] -
Subtract the scores of the 3 excluded students.
[ \text{Sum}_{\text{removed}} = 3 \times 60 = 180 ] [ \text{New Total} = 1360 - 180 = 1180 ] -
Compute the new average with the remaining 17 students.
[ \bar{x}_{17} = \frac{1180}{17} \approx 69.41 ]
Answer: The new average is ≈ 69.41.
4️⃣ Quick Revision Summary
- Average = (\frac{\text{sum of all values}}{\text{number of values}}).
- New average after adding (m) numbers of value (y):
(\displaystyle \bar{x}_{new} = \frac{n\bar{x} + my}{n+m}). - New average after removing (k) numbers of value (z):
(\displaystyle \bar{x}_{new} = \frac{n\bar{x} - kz}{n-k}). - Weighted average of two groups:
(\displaystyle \bar{x}_{total} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}). - Always multiply the old average by its count before adding/subtracting numbers.
- Average of averages uses weights (counts), not just a straight mean.
Tip: When you see “average after adding/removing” or “average of two groups”, think “old sum + (new numbers)” or “weighted sum” – it saves time and avoids mistakes.
Use these rules for all RRB ALP average problems, and you’ll solve them quickly and accurately!