Quantitative Aptitude – Average
For the Nagaland Police Constable Examination
1. Simple Explanation
The average (mean) of a set of numbers is the total sum of the numbers divided by the count of the numbers.
[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} ]
It is also called the arithmetic mean.
2. Key Points
| Point | What it Means | Why It Matters |
|---|---|---|
| Definition | Mean = sum ÷ count | Gives a single value representing the “typical” value. |
| Notation | (\bar{x}) or (A) | Helps in equations and formulas. |
| Basic Formula | (\displaystyle \bar{x} = \frac{x_1+x_2+\dots+x_n}{n}) | Core formula used in all problems. |
| Weighted Average | (\displaystyle \bar{x}_{w} = \frac{\sum w_i x_i}{\sum w_i}) | Used when items have different importance (e.g., marks in subjects). |
| Using Average to Find Missing Value | If average and all other values are known, the missing value can be found by: (\displaystyle x_{\text{missing}} = \bar{x},n - \sum_{\text{known}} x_i) |
Common in exam questions. |
| Changes in Average | Adding a new value changes the average: (\displaystyle \bar{x}{new} = \frac{n\bar{x} + x{\text{new}}}{n+1}) |
Helps in “average after addition” problems. |
| Effect of Outliers | A single very high/low value can skew the average. | Important for interpreting results. |
| Common Pitfalls | • Forgetting to divide by the correct count. • Mixing up weighted and simple averages. • Mis‑reading “average of averages” (needs total sum, not average of averages). |
Leads to wrong answers. |
| Useful Tricks | • When adding equal numbers, the average stays the same. • When adding a number equal to the current average, the average remains unchanged. • For two numbers, average = (\frac{x+y}{2}). |
Saves time. |
3. Worked Example
Problem (Typical for PGI/PGCET style):
The average score of 20 students in a class is 75.
After the exam, the teacher adds the scores of an extra 5 students (who were absent earlier) and the new average becomes 72.
Find the combined average score of those 5 students.
Step‑by‑Step Solution
-
Find the total score of the first 20 students.
[ \text{Total}{20} = \text{Average}{20} \times 20 = 75 \times 20 = 1500 ]
-
Find the total score of all 25 students using the new average.
[ \text{Total}{25} = \text{Average}{25} \times 25 = 72 \times 25 = 1800 ]
-
Find the sum of the 5 new students.
[ \text{Sum}{5} = \text{Total}{25} - \text{Total}_{20} = 1800 - 1500 = 300 ]
-
Compute their average.
[ \text{Average}{5} = \frac{\text{Sum}{5}}{5} = \frac{300}{5} = 60 ]
Answer: The combined average score of the 5 students is 60.
Quick Check
-
Reasonable?
Since the overall average dropped from 75 to 72, the new group must score below 75.
60 is indeed below 75 → ✔️ -
Sum Check:
(20 \times 75 = 1500), (5 \times 60 = 300).
(1500 + 300 = 1800), and (1800 ÷ 25 = 72). ✔️
4. Quick Revision Summary
| Topic | Formula / Idea | Quick Tip |
|---|---|---|
| Mean | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) | Divide the total by the count. |
| Missing Value | (x_{\text{miss}} = \bar{x} n - \sum_{\text{known}}) | “Average × count” gives total. |
| Adding a Value | (\displaystyle \bar{x}{new} = \frac{n\bar{x} + x{\text{new}}}{n+1}) | Keeps track of total easily. |
| Weighted Mean | (\displaystyle \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}) | Use when items have different weights. |
| Average of Averages | Not (\frac{\bar{x}_1 + \bar{x}_2}{2}) unless each group has equal size. | Use total sums. |
| Outlier Effect | One extreme value can change the mean significantly. | Be cautious interpreting averages. |
Final Tips for the Exam
- Read Carefully: Identify whether the problem asks for a simple average or a weighted average.
- Track Totals: Write down the total sum at each step; it often helps in back‑calculations.
- Check Units: If the average is in marks, percentages, etc., keep the units consistent.
- Use Estimation: Quick mental checks (e.g., if the average dropped, the added group must be below the old average) can save time.
- Practice Varied Problems: Work on problems involving “average after adding a number”, “average of averages”, “weighted average”, and “average of a group of numbers with a missing element”.
Good luck with your preparation!