Average
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Important Formulas & Concepts
Study MaterialAverage
Average is one of the most important and scoring topics in Quantitative Aptitude. Questions based on averages are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement examinations.
The concept of average is widely used in mathematics, statistics, business calculations, profit-loss analysis, speed-time-distance problems, and data interpretation.
Why Average is Important?
- Frequently asked in competitive examinations.
- Useful in arithmetic and data interpretation.
- Helps solve speed, age, and partnership problems.
- Important in statistics and business calculations.
- Improves logical and analytical thinking.
What is Average?
Average or Arithmetic Mean is the sum of all observations divided by the total number of observations.
Formula:
Average = Sum of Observations / Number of Observations
| Observation | Value |
|---|---|
| 10 | 10 |
| 15 | 15 |
| 25 | 25 |
| 30 | 30 |
Sum of observations:
10 + 15 + 25 + 30 = 80
Number of observations = 4
Average = 80 / 4 = 20
Basic Formulae of Average
| Concept | Formula |
|---|---|
| Average | Average = Sum / Number of Terms |
| Sum of Observations | Sum = Average × Number of Terms |
| Number of Terms | Number of Terms = Sum / Average |
Properties of Average
1. Average Lies Between Smallest and Largest Observation
The average of a set of observations is always greater than the smallest value and less than the greatest value.
Example:
Average of 3, 7, 9, and 13:
(3 + 7 + 9 + 13) / 4
= 32 / 4
= 8
Clearly:
3 < 8 < 13
2. Equal Observations Have Same Average
If all observations are equal, their average will also be equal to the same value.
Example:
Average of 6, 6, 6, and 6:
(6 + 6 + 6 + 6) / 4
= 24 / 4
= 6
3. Zero is Included in Average
If zero is one of the observations, it must be included while calculating average.
Example:
Average of 3, 6, and 0:
(3 + 6 + 0) / 3
= 9 / 3
= 3
Effects of Changes on Average
| Condition | Effect on Average |
|---|---|
| All numbers increased by a | Average increases by a |
| All numbers decreased by a | Average decreases by a |
| All numbers multiplied by a | Average multiplied by a |
| All numbers divided by a | Average divided by a |
Average of Consecutive Numbers
For consecutive numbers:
Formula:
Average = (First Number + Last Number) / 2
Example:
Find average of numbers from 11 to 21.
Average = (11 + 21) / 2
= 32 / 2
= 16
Average of First n Natural Numbers
Formula:
Average = (n + 1) / 2
| n | Average |
|---|---|
| 10 | 5.5 |
| 20 | 10.5 |
| 50 | 25.5 |
Average of First n Even Numbers
Formula:
Average = n + 1
Example:
Average of first 10 even numbers:
= 10 + 1
= 11
Average of First n Odd Numbers
Formula:
Average = n
Example:
Average of first 9 odd numbers:
= 9
Average of Squares of First n Natural Numbers
Formula:
Average = [(n + 1)(2n + 1)] / 6
Average of Cubes of First n Natural Numbers
Formula:
Average = [n(n + 1)²] / 4
Average of Multiples of a Number
Average of first n multiples of any number:
Formula:
Average = Number × (n + 1) / 2
Example:
Average of first 8 multiples of 5:
= 5 × (8 + 1) / 2
= 45 / 2
= 22.5
Average Speed
Average speed is defined as:
Formula:
Average Speed = Total Distance / Total Time
Average Speed for Equal Distances
If a person travels equal distances at speeds A km/h and B km/h:
Formula:
Average Speed = 2AB / (A + B)
Example:
A person travels equal distances at 25 km/h and 40 km/h.
Average Speed:
= (2 × 25 × 40) / (25 + 40)
= 2000 / 65
≈ 30.77 km/h
Average Speed for Three Equal Distances
If equal distances are travelled at speeds A, B, and C:
Formula:
Average Speed = 3ABC / (AB + BC + CA)
Important Tips on Average
1. Average of Numbers in Progression
If numbers are in arithmetic progression:
- For odd number of terms → Average is middle term.
- For even number of terms → Average is average of two middle terms.
Example:
Average of 3, 6, 9, 12, 15:
Middle term = 9
2. Addition or Removal of Observations
When a new value is added or removed:
- New Sum = Old Sum ± Change
- New Average = New Sum / New Number of Terms
Important Formula Summary
| Concept | Formula |
|---|---|
| Average | Sum / Number of Terms |
| Average of Consecutive Numbers | (First + Last) / 2 |
| Average of First n Natural Numbers | (n + 1) / 2 |
| Average of First n Even Numbers | n + 1 |
| Average of First n Odd Numbers | n |
| Average Speed (Equal Distances) | 2AB / (A + B) |
| Average Speed (Three Equal Distances) | 3ABC / (AB + BC + CA) |
Important Exam Tips
- Memorize all average formulae properly.
- Use middle-term concept for consecutive numbers.
- Be careful while handling addition/removal problems.
- Use harmonic mean formula for equal distance speed problems.
- Practice arithmetic progression-based questions regularly.
- Always verify total observations before calculation.
- Use shortcut methods to save time in exams.
Average is one of the most important topics in Quantitative Aptitude. Strong understanding of formulas, properties, and shortcut methods helps candidates solve aptitude questions quickly and accurately in competitive examinations.