Quantitative Aptitude – Average
(For Arunachal Pradesh Police Constable Examination)
1. Simple Explanation
Average (or mean) is a single value that represents the “central” value of a collection of numbers.
- Arithmetic Mean (AM) – Most common average.
[ \text{AM} = \frac{\text{Sum of all numbers}}{\text{No. of numbers}} ] - Weighted Mean (WM) – Gives different importance (weights) to numbers.
[ \text{WM} = \frac{\sum (w_i \times x_i)}{\sum w_i} ] - Harmonic Mean (HM) – Useful when rates or ratios are involved.
[ \text{HM} = \frac{n}{\sum \frac{1}{x_i}} ] - Geometric Mean (GM) – Useful for growth rates, ratios.
[ \text{GM} = \sqrt[n]{\prod x_i} ]
Key Idea: The average is a representative value that balances out all the data.
2. Key Points
| Concept | Formula | When to Use | Remark |
|---|---|---|---|
| Arithmetic Mean | (\displaystyle \frac{S}{n}) | General data sets, scores, distances, etc. | Easy to calculate. |
| Weighted Mean | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | Data with different importance (e.g., grades, cost). | If all weights equal → Arithmetic Mean. |
| Harmonic Mean | (\displaystyle \frac{n}{\sum \frac{1}{x_i}}) | Rates (speed = distance/time), inverse quantities. | Always ≤ arithmetic mean. |
| Geometric Mean | (\displaystyle \sqrt[n]{\prod x_i}) | Compound growth, percentages. | Always ≤ arithmetic mean. |
| Median | Middle value after sorting | Skewed data, outliers. | Not a “mean” but often confused with it. |
| Mode | Most frequent value | Categorical data. | Not a mean. |
| Relation | AM ≥ GM ≥ HM (for positive numbers) | Helps check calculations. | Equality holds when all numbers are equal. |
Common Tricks
| Scenario | Trick |
|---|---|
| Sum of numbers missing | If you know AM and (n), then (S = \text{AM} \times n). |
| Finding new average after adding a number | |
| [ | |
| \text{New AM} = \frac{(\text{Old AM} \times n) + \text{new number}}{n+1} | |
| ] | |
| Weighted mean with equal totals | If (\sum w_i = 1), then (\text{WM} = \sum w_i x_i). |
| HM of two numbers (a,b) | (\displaystyle \frac{2}{\frac{1}{a} + \frac{1}{b}}) = (\frac{2ab}{a+b}). |
3. Worked Example
Problem (C-1):
In a class, 15 students scored 80, 85, 78, 92, 88, 76, 81, 84, 79, 90, 83, 87, 77, 82, 86.
(a) Find the average score of the class.
(b) If a new student joins and scores 89, what is the new average?
(a) Average of 15 students
- Sum of scores
[ S = 80+85+78+92+88+76+81+84+79+90+83+87+77+82+86 = 1258 ] - Number of students (n = 15).
- Arithmetic Mean
[ \text{AM} = \frac{S}{n} = \frac{1258}{15} \approx 83.87 ] So, average score ≈ 83.9 (to one decimal place).
(b) New average after adding a student scoring 89
- New number of students (n' = 15 + 1 = 16).
- New sum (S' = 1258 + 89 = 1347).
- New AM
[ \text{AM}' = \frac{S'}{n'} = \frac{1347}{16} = 84.1875 ] Rounded to one decimal: 84.2.
Answer: (a) 83.9, (b) 84.2
4. Quick Revision Summary
| Item | Formula | Quick Tip |
|---|---|---|
| Arithmetic Mean | (\displaystyle \frac{\sum x_i}{n}) | Sum → Divide |
| Weighted Mean | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | Multiply by weight first |
| Harmonic Mean | (\displaystyle \frac{n}{\sum \frac{1}{x_i}}) | Inverse sum |
| Geometric Mean | (\displaystyle \sqrt[n]{\prod x_i}) | Product → nth root |
| New AM after adding a number | (\displaystyle \frac{(\text{Old AM} \times n)+x_{\text{new}}}{n+1}) | Use old AM, not total sum |
| AM ≥ GM ≥ HM (for positives) | — | Check reasonableness |
Practice Tips
- Always check units – If data are in kg, m, %, keep consistent.
- Rounded Answers – Police exam usually asks for integer or one decimal; round accordingly.
- Multiple-choice – If only one answer matches the range, use inequalities AM ≥ GM ≥ HM to eliminate.
Good luck with your preparation!