Averages can be defined as the central value in a set of data.
Example:
What is the average of the first five consecutive odd numbers?
Solution:
The first five consecutive odd numbers are: 1, 3, 5, 7, 9.
Here, the number of data or observations is 5 and the sum of these 5 numbers is 25.
So, average = 25 / 5 = 5.
Note:
Sum of 1st n consecutive natural numbers = [n(n+1)]/2
Average of 1st n consecutive natural numbers = (n+1)/2
Average of first n natural numbers  (n+1) / 2 
Average of squares of first n natural numbers  (n+1) (2n+1) / 6 
Average of cubes of first n natural numbers  n (n+1)^{2} / 4 
Average of first n even numbers  n + 1 
Average of squares of first n even numbers  2 (n+1) (2n+1) / 3 
Average of cube of first n even numbers  2n (n+1)^{2} 
Average of first n odd numbers  n 
Average of squares of first n odd numbers  (2n+1) (2n1) / 3 
Average of cube of first n odd numbers  n(2n^{2} – 1) 
Example 1:
The average of a batsman in 16 innings is 36. In the next innings, he is scoring 70 runs. What will be his new average?
Solution:
Conventionally solving:
New average = (old sum+ new score)/(total number of innings) = ((16 ×36)+70)/((16+1)) = 38
Shortcut technique:
Example 2:
The average mark of 19 children in a particular school is 50. When a new student with marks 75 joins the class, what will be the new average of the class?
Solution:
Step 1) Take the difference between the old average and the new marks = 7550=25
Step 2) This score of 25 is distributed over 20 students => 25/20 = 1.25
Step 3) Hence, the average increases by 1.25=> 50+1.25 = 51.25.
Example 3:
The average age of Mr. Mark’s 3 children is 8 years. A new baby is born. Find the average age of all his children?
Solution:
The new age will be 0 years. The difference between the old average and the new age = 08= 8
This age of 8 years is spread over 4 children => (8/4= 2) Hence, the average reduces to 82= 6 years.
Example 1:
The average age of 29 students is 18. If the age of the teacher is also included the average age of the class becomes 18.2. Find the age of the teacher?
Solution:
Conventionally solving:
Let the average age of the teacher = x(29 × 18 + x × 1)/30Solving for x, we get x = 24.
Shortcut Technique:
Using the shortcut, based on the same method used previously:
Step 1: Calculate the change in average = 18.2 – 18 = 0.2.This change in 0.2 is reflected over a sample size of 30.
The new age is increased by 30 × 0.2 = 6 years above the average i.e. 18 + 6 = 24; which is the age of the teacher.
The concept of an assumed mean is not new. It is widely used to reduce the calculation in finding the average in statistics where the data is huge.
Here, We will demonstrate the application of the assumed mean to solve some aptitude questions based on averages and weighted averages.
Let us take an example to understand the concept
Example: In a class of 30 students, the average age is 12 years. If the age of the class teacher is included, the new average age of the class becomes 13 years. Find the age of the class teacher.
Solution:
Standard Approach
Applying the standard approach, the total age of the 30 students = years. When the class teacher is included, the new total age of the class = years. Note that the increase in the total age is because of the class teacher only. Hence the age of the class teacher = 403 – 360 = 43 years
Deviation method
To understand the deviation method, let us simulate the problem. In the case of average age, assume that each student has 12 chocolates with them.
Therefore, he came with 30 + 13 = 43 chocolates. Or the age of the class teacher is 43 years.
The weighted arithmetic mean, usually denoted by
,
where x1, x2, x3, …, xn are averages and w1, w2, w3, .., wn are their respective weightages
Example: In a class of 25 boys and 15 girls, the average heights of the two groups of boys and girls are 150 cm and 140 cm respectively. Find the average height of the class.
Solution:
Standard approach:
Total weight of the group of boys = and for the group of girls, the total weight =Therefore, the average height of the class =
Deviation Approach:
Assume that each boy and each girl is carrying 150 and 140 chocolates respectively.
Q1: The average goal scored by 15 selected players in EPL is 16. The maximum number of goals scored by a player is 20 and the minimum is 12. The goals scored by players are between 12 and 20. What can be the maximum number of players who scored at least 18 goals?
a) 10
b) 5
c) 9
d) 6
e) None of these
Solution: Option (c)
To maximize the number of players who scored 18 and above number of goals, one should assume that only one person has scored 20. To counter him, there will be one person who will score 12 goals.
i.e. 15 – 2 = 13 players left.
Now to maximize the 18 and above goals for every two players who are scoring 18, there will be one player scoring 12. This is done, to arrive at an average of 16. We will have 8 players with a score of 18 and 4 players with a score of 12. The last player will have a score of 16 Thus, the maximum number of people with 18 and more goals = 9.
Q2: The average weight of a group of 8 girls is 50 kg. If 2 girls R and S replace P and Q, the new average weight becomes 48 kg. The weight of P= Weight of Q and the weight of R = Weight of another girl T is included in the group and the new average weight becomes 48 kg. Weight of T= Weight of R. Find the weight of P?
a) 48 kgs
b) 52 kgs
c) 46 kgs
d) 56 kgs
Solution: Option (d)
8 x 50 +R+SPQ= 48×8 R+SPQ=16
P+QRS= 16 R=S and P=Q
PR=8
One more person is included and the weight = 48 kg
Let the weight be a = (48 × 8 + a)9/9 = 48
A = 48 kg= weight of R
=> Weight of P= 48+8= 56 kg.
Q3: The average number of goals scored by 15 selected players in EPL is 16. The maximum number of goals scored by a player is 20 and the minimum is 12. The goals scored by players is between 12 and 20. What can be the maximum number of players who scored at least 18 goals?
a) 10
b) 5
c) 9
d) 6
Solution: Option (c)
To maximize the number of players who scored 18 goals or more, one should assume that only one person has scored 20. To counter him, there will be one person who will score 12 goals.
i.e. 15 – 2 = 13 players left.
Now to maximize the 18 and above goals for every two players who are scoring 18, there will be one player scoring 12. This is done, to arrive at an average of 16. We will have 8 players with a score of 18 and 4 players with a score of 12. The last player will have a score of 16 Thus, the maximum number of people with 18 and more goals = 9.
194 videos156 docs192 tests

Detailed Notes: Averages Doc  4 pages 
Practice Questions: Averages Doc  8 pages 
Overview: Averages Video  01:56 min 
1. What are averages? 
2. What is the formula to find the average? 
3. What is the difference between median and mode? 
4. How can deviation be used to find the average? 
5. What is the concept of weighted mean/average? 
194 videos156 docs192 tests

Detailed Notes: Averages Doc  4 pages 
Practice Questions: Averages Doc  8 pages 
Overview: Averages Video  01:56 min 

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