Kerala Police Sub‑Inspector (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 calculated by adding all the values and dividing by the number of values.
Formula
[ \text{Average} = \frac{\displaystyle\sum_{i=1}^{n} x_i}{n} ] where
(x_i) – individual values
(n) – total number of values
1. Simple Explanation
- Add all the numbers in the set.
- Count how many numbers there are.
- Divide the sum by the count.
Example:
Numbers: 4, 8, 12, 16
Sum = 4+8+12+16 = 40
Count = 4
Average = 40 ÷ 4 = 10
2. Key Points for SI Examination
| Aspect | What to Remember |
|---|---|
| Definition | Mean of (n) numbers = total sum ÷ (n). |
| Typical Problems | • Average of a set. • Finding missing value when average and some values are given. • Average after adding new data. • Weighted average (different weights). |
| Formula Variations | • If two averages are known and a common element is added: [ |
\frac{n_1A_1 + n_2A_2}{n_1+n_2} ] | | Common Mistakes | • Mixing up total sum with average.
• Forgetting to divide by the correct count.
• Not converting units (e.g., averages in minutes vs hours). | | Useful Tips | • Always check the units.
• Use “back‑calculation” when a missing number is involved.
• For weighted averages, treat each weight as the “count” for that value. |
3. Worked Example
Problem
A batch of 5 machines produced the following output (in units) in a day: 120, 150, 180, 210, and (x) units.
The average output of all 5 machines is 190 units.
Find the output of the fifth machine.
Step‑by‑Step Solution
-
Write the average formula for all 5 machines
[ \frac{120 + 150 + 180 + 210 + x}{5} = 190 ] -
Multiply both sides by 5 to eliminate the denominator
[ 120 + 150 + 180 + 210 + x = 190 \times 5 ] [ 120 + 150 + 180 + 210 + x = 950 ] -
Sum the known values
[ 120 + 150 = 270 \ 270 + 180 = 450 \ 450 + 210 = 660 ] -
Solve for (x)
[ 660 + x = 950 \ x = 950 - 660 \ x = 290 ] -
Answer
The fifth machine produced 290 units.
4. Quick Revision Summary
| Item | What to Do |
|---|---|
| Add all numbers | Use a calculator or mental addition. |
| Count the numbers | Don’t forget any value; double‑check the count. |
| Divide | Sum ÷ count = average. |
| Missing value | Multiply the given average by the count, subtract the known sum, solve for the unknown. |
| Weighted average | Treat each weight as a repetition of the value. |
| Check units | Ensure consistency (e.g., hours vs minutes). |
Tip for the exam:
In multiple‑choice questions, if you’re unsure, try a quick “back‑calculation” – estimate the sum first, then see if any answer choice matches the average you’d get.
Practice Question
The average height of 6 children is 140 cm. Four of them are 150 cm, 155 cm, 145 cm, and 149 cm. Find the sum of the heights of the remaining two children.
Solution (quick):
Total sum = 140 × 6 = 840 cm.
Known sum = 150+155+145+149 = 599 cm.
Remaining sum = 840 – 599 = 241 cm.
(The two heights together must add to 241 cm.)
Use this framework for all average‑related problems in the quantitative aptitude section. Happy studying!