📚 Quantitative Aptitude – Average
Average (Mean) is the sum of a set of numbers divided by the count of numbers.
It gives a central value that represents the whole data set.
1. Simple Explanation
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Arithmetic Mean (Unweighted Average)
[ \text{Mean} = \frac{\sum \text{values}}{\text{number of values}} ] -
Weighted Average – each value has a weight (importance)
[ \text{Weighted Mean} = \frac{\sum (\text{value} \times \text{weight})}{\sum \text{weights}} ] -
Harmonic Mean – used when averaging rates or ratios
[ \text{Harmonic Mean} = \frac{n}{\sum \frac{1}{\text{value}}} ]
Why it matters: Averages simplify data, help compare groups, and are used in many JEE‑type problems (average speeds, scores, weights, etc.).
2. Key Points
| Concept | Formula | When to Use |
|---|---|---|
| Arithmetic Mean | (\displaystyle \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}) | Equal weight, ordinary averages |
| Weighted Mean | (\displaystyle \bar{x}w = \frac{\sum{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}) | Different importance (e.g., marks in subjects, speeds over distances) |
| Harmonic Mean | (\displaystyle H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}) | Averaging rates (speed, density) |
| Median | Middle value in an ordered set | Robust against outliers |
| Mode | Most frequent value | Categorical data |
| Properties | • Non‑negative if all numbers non‑negative. • Mean ≤ Max, Mean ≥ Min. • Adding a number changes the mean in a predictable way. |
• Helps estimate totals or missing data. • Useful in solving “average change” type problems. |
3. Worked Example
Problem (Typical JNU Entrance style):
The average score of 5 students in a test is 70. Three of them score 65, 75, and 80. What did the remaining two students score?
Steps
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Let the unknown scores be (x) and (y).
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Set up the equation using the definition of average:
[ \frac{65 + 75 + 80 + x + y}{5} = 70 ]
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Multiply both sides by 5:
[ 65 + 75 + 80 + x + y = 350 ]
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Sum the known scores:
[ 220 + x + y = 350 ]
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Isolate (x + y):
[ x + y = 350 - 220 = 130 ]
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Interpretation:
The sum of the remaining two scores must be 130.
Any pair of numbers that sum to 130 works (e.g., 60 & 70, 65 & 65, etc.)
(If the problem asks for “the possible scores”, you’d list possibilities or note that the information is insufficient for unique values.)
Another Example – Weighted Average (Average Speed)
Problem:
A train travels 60 km at 60 km/h and then 40 km at 80 km/h. Find the average speed for the whole journey.
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Use weighted mean with distance as weight:
[ \bar{v} = \frac{60 \times 60 + 40 \times 80}{60 + 40} = \frac{3600 + 3200}{100} = \frac{6800}{100} = 68\ \text{km/h} ]
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Answer:
The average speed is 68 km/h.
4. Quick Revision Summary
- Arithmetic Mean = total ÷ count.
- Weighted Mean = Σ(weight × value) ÷ Σ(weight).
- Harmonic Mean = count ÷ Σ(1/value).
- Adding a new value:
[ \bar{x}{new} = \frac{n\bar{x}{old} + x_{\text{new}}}{n+1} ] - Removing a value:
[ \bar{x}{new} = \frac{n\bar{x}{old} - x_{\text{removed}}}{n-1} ] - Average problems often involve:
• Missing data estimation.
• Rate problems (speed, density).
• Comparing groups. - Tips for JNU Entrance:
- Read carefully – check if it's a weighted scenario.
- Set up an equation using the definition of average.
- Simplify arithmetic – use mental math for sums.
- Check units – especially in speed or rate problems.
Practice Idea:
- Find the average of 8 numbers if the average is 15 and the sum of 4 numbers is 50.
- A student scored 45, 55, 65, 75, and 85 in 5 subjects with weightages 1, 2, 3, 4, 5 respectively. Compute the weighted average.
Use these notes to master the concept of averages and tackle a wide variety of quantitative problems confidently!