Episode 8 — Aptitude and Reasoning / 8.8 — Average
8.8.a Concepts and Formulas
1. What Is an Average?
An average (or arithmetic mean) is a single number that represents the central value of a set of numbers. It is the value each item would have if the total were distributed equally among all items.
Average = Sum of all observations / Number of observations
Or, written symbolically:
A = S / n
Where:
A = Average
S = Sum of all observations
n = Number of observations (count)
Three rearrangements:
Sum = Average x Count S = A x n
Count = Sum / Average n = S / A
Average = Sum / Count A = S / n
Worked Example:
Find the average of 12, 15, 18, 21, and 24.
Sum = 12 + 15 + 18 + 21 + 24 = 90
Count = 5
Average = 90 / 5 = 18
2. Sum from Average
This reverse application is tested frequently. If you know the average and the count, you can find the total sum.
Sum = Average x Count
Worked Example:
The average marks of 40 students is 72. What is the total marks of all students?
Total marks = Average x Count
= 72 x 40
= 2,880
3. Weighted Average
When different groups have different sizes (weights), a simple average does not work. We need a weighted average.
Weighted Average = (w1*x1 + w2*x2 + ... + wn*xn) / (w1 + w2 + ... + wn)
Where:
x1, x2, ..., xn = values (e.g., averages of each group)
w1, w2, ..., wn = weights (e.g., number of items in each group)
Worked Example:
In a class, 30 boys have an average score of 70 and 20 girls have an average score of 80. Find the average score of the entire class.
Weighted Average = (30 x 70 + 20 x 80) / (30 + 20)
= (2100 + 1600) / 50
= 3700 / 50
= 74
Key insight: The combined average (74) is NOT the simple average of 70 and 80 (which would be 75). It is closer to 70 because the group with average 70 has more members.
4. Average of Consecutive Numbers
4.1 Consecutive Natural Numbers from a to b
Average = (a + b) / 2 (i.e., the middle value)
Sum = n x (a + b) / 2
Count n = (b - a) + 1
Worked Example:
Find the average of all integers from 15 to 45.
Average = (15 + 45) / 2 = 60 / 2 = 30
Count = (45 - 15) + 1 = 31
Sum = 31 x 30 = 930
4.2 First n Natural Numbers (1, 2, 3, ..., n)
Sum = n(n + 1) / 2
Average = (n + 1) / 2
Worked Example:
Find the average of the first 50 natural numbers.
Average = (50 + 1) / 2 = 51 / 2 = 25.5
4.3 First n Even Natural Numbers (2, 4, 6, ..., 2n)
Sum = n(n + 1)
Average = (n + 1)
Worked Example:
Find the average of the first 20 even natural numbers.
Average = 20 + 1 = 21
Verification: First 20 even numbers: 2, 4, 6, ..., 40. Average = (2 + 40)/2 = 21. Correct.
4.4 First n Odd Natural Numbers (1, 3, 5, ..., 2n-1)
Sum = n^2
Average = n
Worked Example:
Find the average of the first 15 odd natural numbers.
Average = 15
Verification: First 15 odd numbers: 1, 3, 5, ..., 29. Average = (1 + 29)/2 = 15. Correct.
4.5 Consecutive Numbers in Arithmetic Progression
For any AP with first term a, common difference d, and n terms:
Average = a + (n - 1) x d / 2
= (First term + Last term) / 2
5. Average Speed
This is one of the most commonly tested -- and most commonly mistaken -- concepts.
5.1 Same Distance, Different Speeds
If a person travels the same distance at two different speeds S1 and S2:
Average Speed = 2 x S1 x S2 / (S1 + S2)
This is the harmonic mean, NOT the arithmetic mean.
Worked Example:
A car travels from A to B at 40 km/h and returns at 60 km/h. Find the average speed for the entire journey.
Average Speed = 2 x 40 x 60 / (40 + 60)
= 4800 / 100
= 48 km/h
Common mistake: (40 + 60) / 2 = 50 km/h is WRONG.
5.2 Same Time, Different Speeds
If a person travels for the same time at two different speeds:
Average Speed = (S1 + S2) / 2
In this case, the simple arithmetic mean IS correct.
Worked Example:
A person walks at 4 km/h for 2 hours and then at 6 km/h for 2 hours. Find the average speed.
Average Speed = (4 + 6) / 2 = 5 km/h
Verification:
Distance 1 = 4 x 2 = 8 km
Distance 2 = 6 x 2 = 12 km
Total distance = 20 km
Total time = 4 hours
Average speed = 20 / 4 = 5 km/h. Correct.
5.3 Three Different Speeds, Same Distance
Average Speed = 3 x S1 x S2 x S3 / (S1*S2 + S2*S3 + S3*S1)
5.4 General Formula
In the most general case:
Average Speed = Total Distance / Total Time
Always fall back to this when confused.
6. Effect of Adding or Removing an Element
6.1 Adding a New Element
If the average of n numbers is A, and a new number x is added:
New Average = (nA + x) / (n + 1)
Worked Example:
The average of 5 numbers is 20. If a new number 32 is added, what is the new average?
Old sum = 5 x 20 = 100
New sum = 100 + 32 = 132
New average = 132 / 6 = 22
6.2 Removing an Element
If the average of n numbers is A, and a number x is removed:
New Average = (nA - x) / (n - 1)
Worked Example:
The average of 8 numbers is 25. If the number 17 is removed, what is the new average of the remaining 7 numbers?
Old sum = 8 x 25 = 200
New sum = 200 - 17 = 183
New average = 183 / 7 = 26.14 (approx)
6.3 Replacing an Element
If one number is replaced by another, the change in average tells us the relationship.
Change in sum = New element - Old element
Change in average = Change in sum / n
Worked Example:
The average of 10 numbers is 15. If one number 24 is replaced by 12, what is the new average?
Change in sum = 12 - 24 = -12
Change in average = -12 / 10 = -1.2
New average = 15 - 1.2 = 13.8
6.4 Finding the Missing Number
A very common exam pattern: given the new average after adding/removing one element, find that element.
Worked Example:
The average age of 4 friends is 28 years. A fifth friend joins them and the average becomes 30. What is the age of the fifth friend?
Old sum = 4 x 28 = 112
New sum = 5 x 30 = 150
Age of 5th friend = 150 - 112 = 38 years
7. Combined Average of Two or More Groups
When two (or more) groups with known averages are combined:
Combined Average = (n1 x A1 + n2 x A2) / (n1 + n2)
Where:
n1, n2 = sizes of the two groups
A1, A2 = averages of the two groups
For three groups:
Combined Average = (n1*A1 + n2*A2 + n3*A3) / (n1 + n2 + n3)
Worked Example:
Section A has 40 students with an average of 65 marks. Section B has 35 students with an average of 70 marks. Find the combined average.
Combined Average = (40 x 65 + 35 x 70) / (40 + 35)
= (2600 + 2450) / 75
= 5050 / 75
= 67.33
Important property: The combined average always lies between the two individual averages.
A1 <= Combined Average <= A2 (assuming A1 <= A2)
The combined average is closer to the average of the larger group.
8. Effect on Average When Every Element Changes
8.1 Each Element Increased by a Constant k
New Average = Old Average + k
8.2 Each Element Decreased by a Constant k
New Average = Old Average - k
8.3 Each Element Multiplied by a Constant k
New Average = Old Average x k
8.4 Each Element Divided by a Constant k
New Average = Old Average / k
Worked Example:
The average salary of 50 employees is Rs 18,000. If every employee gets a raise of Rs 2,500, what is the new average?
New Average = 18,000 + 2,500 = Rs 20,500
Worked Example:
The average of 6 numbers is 30. If each number is doubled, what is the new average?
New Average = 30 x 2 = 60
9. Age-Based Average Problems
These are extremely popular in exams. The key ideas:
9.1 After t Years
If the current average age of n persons is A, then after t years:
New Average = A + t
Because each person's age increases by t, so the sum increases by n*t, and the average increases by t.
9.2 Before t Years
Average t years ago = A - t
9.3 Birth and Death in a Group
When a new person is born into a group or someone dies, both the count and sum change.
Worked Example:
The average age of a family of 4 members is 25 years. A baby is born. What is the new average age of the family?
Old sum = 4 x 25 = 100
New sum = 100 + 0 = 100 (baby's age = 0)
New average = 100 / 5 = 20 years
Worked Example:
5 years ago, the average age of a family of 4 was 30 years. A baby was born 3 years ago. What is the current average age of the 5-member family?
5 years ago, sum of ages of 4 members = 4 x 30 = 120
Current sum of those 4 members = 120 + (4 x 5) = 120 + 20 = 140
Baby's current age = 3 years
Total current sum = 140 + 3 = 143
Current average = 143 / 5 = 28.6 years
10. Batting / Bowling Average (Cricket Problems)
10.1 Batting Average
Batting Average = Total runs scored / Number of innings played
10.2 When Average Increases After an Innings
If after playing n innings, a batsman's average increases by x after scoring R runs in the (n+1)th innings:
R = New Average + n x (Increase in average)
R = (Old Average + x) + n*x
R = Old Average + (n + 1) x x
Or simply:
Old average after n innings = A
New average after (n+1) innings = A + x
Runs in (n+1)th innings = (n+1)(A + x) - nA = A + (n+1)x
Worked Example:
A batsman has an average of 40 runs after 20 innings. After the 21st innings, his average increases by 2. How many runs did he score in the 21st innings?
Old sum = 20 x 40 = 800
New average = 40 + 2 = 42
New sum = 21 x 42 = 882
Runs in 21st innings = 882 - 800 = 82
11. Median, Mean, and Mode -- Basics
While Average (Mean) is the primary focus, understanding how it relates to Median and Mode is important.
11.1 Mean (Arithmetic Mean)
Mean = Sum of all values / Number of values
This is what we have been studying throughout this section.
11.2 Median
The middle value when data is arranged in ascending or descending order.
For odd n: Median = value at position (n + 1) / 2
For even n: Median = average of values at positions n/2 and (n/2 + 1)
Worked Example:
Find the median of: 3, 7, 2, 9, 5
Arranged in order: 2, 3, 5, 7, 9
n = 5 (odd)
Median = value at position (5 + 1)/2 = 3rd position = 5
Worked Example:
Find the median of: 4, 8, 1, 6, 3, 9
Arranged in order: 1, 3, 4, 6, 8, 9
n = 6 (even)
Median = (4 + 6) / 2 = 5
11.3 Mode
The value that appears most frequently in a data set.
Mode = the observation with the highest frequency
Worked Example:
Find the mode of: 3, 5, 7, 3, 8, 3, 9, 5
3 appears 3 times (most frequent)
Mode = 3
11.4 Relationship Between Mean, Median, and Mode
For a moderately skewed distribution (an empirical approximation):
Mode = 3 x Median - 2 x Mean
This is not exact but is useful for competitive exams.
11.5 Key Differences
| Property | Mean | Median | Mode |
|---------------|----------------|----------------|----------------|
| Uses all data | Yes | No | No |
| Affected by | Yes | No | No |
| outliers | | | |
| Uniqueness | Always unique | Always unique | Can be multiple|
| Best for | Symmetric data | Skewed data | Categorical |
12. Average of Squared and Cubed Numbers
12.1 Average of Squares of First n Natural Numbers
Sum of squares = n(n + 1)(2n + 1) / 6
Average of squares = (n + 1)(2n + 1) / 6
Worked Example:
Find the average of squares of the first 10 natural numbers.
Average = (10 + 1)(2 x 10 + 1) / 6
= 11 x 21 / 6
= 231 / 6
= 38.5
12.2 Average of Cubes of First n Natural Numbers
Sum of cubes = [n(n + 1) / 2]^2
Average of cubes = n(n + 1)^2 / 4
Worked Example:
Find the average of cubes of the first 6 natural numbers.
Average = 6 x (6 + 1)^2 / 4
= 6 x 49 / 4
= 294 / 4
= 73.5
13. Average in Data Interpretation (Grouped Data)
When data is given in a frequency table:
Average = Sum of (frequency x value) / Sum of frequencies
= (f1*x1 + f2*x2 + ... + fn*xn) / (f1 + f2 + ... + fn)
For class intervals, use the mid-value of each class as the representative value.
Mid-value = (Lower limit + Upper limit) / 2
Worked Example:
Marks 10-20 20-30 30-40 40-50 Students 5 8 12 5
Mid-values: 15, 25, 35, 45
Average = (5x15 + 8x25 + 12x35 + 5x45) / (5 + 8 + 12 + 5)
= (75 + 200 + 420 + 225) / 30
= 920 / 30
= 30.67
Summary of All Formulas
1. Average = Sum / Count
2. Sum = Average x Count
3. Weighted Average = Sum(wi * xi) / Sum(wi)
4. Avg of first n natural numbers = (n + 1) / 2
5. Avg of first n even natural numbers = n + 1
6. Avg of first n odd natural numbers = n
7. Avg of consecutive a to b = (a + b) / 2
8. Avg speed (same distance) = 2*S1*S2 / (S1 + S2)
9. Avg speed (same time) = (S1 + S2) / 2
10. New avg (element added) = (nA + x) / (n + 1)
11. New avg (element removed) = (nA - x) / (n - 1)
12. Combined avg of groups = (n1*A1 + n2*A2) / (n1 + n2)
13. Each element + k => New avg = Old avg + k
14. Each element x k => New avg = Old avg x k
15. After t years, avg age = Current avg + t
16. Batting avg = Total runs / Innings
17. Avg of squares (1 to n) = (n+1)(2n+1) / 6
18. Avg of cubes (1 to n) = n(n+1)^2 / 4
19. Median (odd n) = middle value
20. Mode = most frequent value