Karnataka Police SI – Quantitative Aptitude
Topic: Average
Average (also called mean) is a measure of central tendency that represents the “typical” value of a set of numbers.
It is obtained by adding all the values and dividing by the count of values.
1. Simple Explanation
| Concept | What It Means | Formula |
|---|---|---|
| Arithmetic Mean | The usual “average” – sum of all items ÷ number of items | (\displaystyle \bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}) |
| Weighted Average | Items have different importance (weights) | (\displaystyle \bar{x}=\frac{\sum_{i=1}^{n}w_i x_i}{\sum_{i=1}^{n}w_i}) |
| Combined Average of Groups | Two or more groups with known averages and sizes | (\displaystyle \bar{x}_{\text{total}}=\frac{n_1\bar{x}_1+n_2\bar{x}_2+\dots}{n_1+n_2+\dots}) |
Key Point – In all cases the average is sum of weighted values divided by sum of weights (or just the count when all weights are 1).
2. Key Points (for quick reference)
| # | Topic | Important Notes |
|---|---|---|
| 1 | Arithmetic Mean | • (\bar{x}= \frac{S}{n}) where (S) = sum of data, (n) = number of data points. • Used in most SI exam questions. |
| 2 | Weighted Average | • Weights can be frequency, importance, or any numeric factor. • Example: Speed = (\frac{50 km \times 2 h + 30 km \times 1 h}{2+1}). |
| 3 | Average of Groups | • If two groups have averages (A_1, A_2) and sizes (n_1, n_2):\n (\displaystyle \bar{A}=\frac{n_1A_1+n_2A_2}{n_1+n_2}). |
| 4 | Average after Adding a New Value | • New average (\bar{A}{new}=\frac{n\bar{A}+x{new}}{n+1}). |
| 5 | Average after Removing a Value | • Remove (x):(\displaystyle \bar{A}_{new}=\frac{n\bar{A}-x}{n-1}). |
| 6 | Average with Increase/Decrease | • If each value increases by (k): new average = old average + (k). |
| 7 | Finding Sum from Average | • (S = n \times \bar{x}). |
| 8 | Finding Count from Sum and Average | • (n = \frac{S}{\bar{x}}). |
| 9 | Common Mistakes | • Mixing up “mean” and “median”. • Forgetting to divide by the correct count. • Using wrong weights. |
3. Worked Example
Problem
The average score of 12 students in a class is 78.
Three students scored 100 each.
Find the average score of the remaining 9 students.
Step‑by‑Step Solution
-
Find total score of all 12 students
(\displaystyle S_{\text{total}} = \bar{S}\times n = 78 \times 12 = 936) -
Find total score of the 3 students who scored 100
(\displaystyle S_{100} = 3 \times 100 = 300) -
Compute total score of remaining 9 students
(\displaystyle S_{\text{remaining}} = S_{\text{total}} - S_{100} = 936 - 300 = 636) -
Compute their average
(\displaystyle \bar{S}{\text{remaining}} = \frac{S{\text{remaining}}}{9} = \frac{636}{9} = 70.666\ldots)Usually we give the answer to one decimal place or nearest integer:
(\boxed{70.7}) (or 71, depending on the exam’s rounding rule).
Quick Practice
Q: The average of 5 numbers is 20.
If one number is 12, what is the average of the remaining 4 numbers?
A:
[ S_{\text{total}} = 5 \times 20 = 100\ S_{\text{remaining}} = 100 - 12 = 88\ \bar{S}_{\text{remaining}}