📚 Study Notes – Average
Madhya Pradesh PSC – Quantitative Aptitude
The concept of average (also called mean) is a basic tool used in many PSC exams to solve questions on distribution, marks, wages, speeds, etc.
Mastery of averages will help you save time and reduce errors in multiple-choice questions.
1. Simple Explanation
| Term | Definition | Example |
|---|---|---|
| Arithmetic Mean (Average) | Sum of all values ÷ Number of values | For marks 45, 55, 60, 70: (45+55+60+70)/4 = 60 |
| Weighted Average | (Sum of (value × weight)) ÷ (Sum of weights) | If 30% of marks are in Maths (score 80) and 70% in Science (score 70): (80×30 + 70×70)/(30+70) = 73 |
- Why use averages?
- Quickly summarises large data sets.
- Helps in comparing performances, speeds, wages, etc.
- Often used in average speed problems, average wages problems, and average marks questions.
2. Key Points to Remember
| Concept | Important Formula | Tips |
|---|---|---|
| Arithmetic Mean | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) | n = total items. Avoid dividing by 0. |
| Weighted Mean | (\displaystyle \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}) | Weights are percentages or any relative importance. |
| Changing a Data Point | If one value changes, update sum then divide. | Keep a running total to save time. |
| Average Speed | (\displaystyle \text{Avg. Speed} = \frac{\text{Total Distance}}{\text{Total Time}}) | Use consistent units (km, h). |
| Average Wage | (\displaystyle \text{Avg. Pay} = \frac{\text{Total Earnings}}{\text{Hours Worked}}) | Useful for overtime, part‑time workers. |
| Average of Two Numbers | (\displaystyle \frac{a+b}{2}) | Quick mental trick: Find midpoint. |
| Average of Three Numbers | (\displaystyle \frac{a+b+c}{3}) | Often used in “average of 3 scores” questions. |
| Average of N Numbers | Same as arithmetic mean. | If N is huge, use sum of series formulas. |
Common Mistakes
- Mixing units – Always convert to common unit before dividing.
- Incorrect denominator – For weighted average, denominator is sum of weights, not total items.
- Rounding too early – Keep decimals until final answer.
- Forgetting to add all values – Double‑check the sum.
Quick Fix for Time Pressure
- Use “half‑sum” technique for two numbers: add them, halve.
- Use “average of averages”: If you know averages of groups, compute weighted average.
3. Worked Example
Question (Typical MCQ):
The average marks of 40 students in a class is 75. After an additional test, the average marks of the same 40 students rises to 78. What is the total marks obtained by all students in the new test?
Solution Steps
-
Find total marks before the new test
[ \text{Total}_1 = \text{Average}_1 \times \text{Number of students} = 75 \times 40 = 3000 ] -
Find total marks after the new test
[ \text{Total}_2 = \text{Average}_2 \times 40 = 78 \times 40 = 3120 ] -
Difference gives total marks in new test
[ \text{Marks in new test} = \text{Total}_2 - \text{Total}_1 = 3120 - 3000 = 120 ]
Answer: 120 marks (total) or 3 marks per student on average.
4. Quick Revision Summary
| Topic | Formula | Typical Use | Quick Tip |
|---|---|---|---|
| Arithmetic Mean | (\displaystyle \bar{x}=\frac{\sum x_i}{n}) | Marks, wages, speeds | Keep running sum |
| Weighted Mean | (\displaystyle \bar{x}_w=\frac{\sum w_i x_i}{\sum w_i}) | Curriculum weights, mixed exams | Denominator = sum of weights |
| Average Speed | (\displaystyle \frac{\text{Distance}}{\text{Time}}) | Travel, transport | Same units |
| Average Wage | (\displaystyle \frac{\text{Earnings}}{\text{Hours}}) | Salary, overtime | Convert to same currency/time |
| Average of Two Numbers | (\displaystyle \frac{a+b}{2}) | Quick checks | Half‑sum trick |
| Average of Three Numbers | (\displaystyle \frac{a+b+c}{3}) | Small set | Use 1/3 of sum |
- Practice Tip: Solve at least 5–10 average problems every day.
- Time Saver: For large n, use formulas for arithmetic series or known sums.
- Check Work: Always re‑calculate the denominator and make sure units match.
Good Luck!
Master these basics, and you’ll handle most average‑related questions in the Madhya Pradesh PSC Quantitative Aptitude section with confidence.