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Average

Traditionally, average is calculated by dividing the sum of all the numbers by the number of numbers.
2366.png
 
For example, the average of the four numbers 214, 215, 219, 224 will be
2375.png

Central Value Meaning of Average

Average can also be seen as the central value of all the given values.

Applying this definition for the above example let us assume the central value of all the given numbers = 214
 
Now, find the deviations of all the numbers from 214
214 215 219 224
 
When assumed central value is (214), the sum of the deviations
= 0 + 1 + 5 + 10
 
Now, finding the average of deviations gives us
2384.png
 
So, average = assumed central value + average of deviations = 214 + 4 = 218
 
Thus, we can assume any value to be the assumed average and then find the average of all the deviations; and when we add all the numbers and divide it by number of numbers, 0 is assumed to be the central value.
 
Example-1
Average age of A, B and C is 84 years. When D joins them the average age of A, B, C and D becomes 80 years. A new person E, whose age is 4 years more than D, replaces A and the average age of B, C, D and E becomes 78 years. What is the age of A?
Solution
Since the average age of A, B and C is 84 years so, we can assume that age of A, B and C is 84 years.
 
A = 84 years
 
B = 84 years
 
C = 84 years
 
After D has joined them,
 
  Initially Finally
A 84 years 80 years
B 84 years 80 years
C 84 years 80 years
D ….. 80 years
 
Decrease in the age of A, B and C can be attributed to the increase in the age of D. So, after getting 12 years in total (4 years each from A, B and C) D is at 80 years. The original age of D = 80 − 12 = 68 years.
 
So, age of E = 72 years
 
Now, the average age of A, B, C and D = 80 years; 
A + B + C + D = 320
 
And average of B, C, D and E = 78 years; B + C + D + E = 312
(Since the average difference between the age of A and E is 2 years,)
 
Difference (A − E) = 2 × 4 = 8 years
 
Since E = 72 years, so A = 80 years
 
By using central value method of averages every question of average can be done by mental calculation only.
 
 
Example-2
Average of ten two digit numbers is S. However when we reverse one of the numbers AB as BA from the given 10 numbers, then the average becomes S + 1.8. What is the value of B − A?
Solution
Average of 10 numbers is increasing by 1.8, so it can be assumed that 1.8 has been added to all the numbers.
 
So, BA is 1.8 × 10 = 18 more than AB.
 
There are so many two-digit numbers which satisfy above condition. Using hit and trial method, the number can be 13, 24, 35, 46, 57, 68, 79. In every case, difference between the digits = 2
 
Otherwise, we can use the formula (BA − AB) = 9 × (B − A)
 
Where BA and AB are two-digit numbers.
 
 
Example-3
The average score of Rahul Dravid after 25 innings is 46 runs per innings. If after the 26th innings, his average runs increased by 2 runs, then what is his score in the 26th inning?
Solution
Runs in 26th inning = Total runs after 26th innings − Total runs after 25th innings
 
= 26 × 48 − 25 × 46 = 98
 
Alternatively, this question can be done by the above given central value meaning of average. Since the average increases by 2 runs per innings, we can assume that 2 runs have been added to his score in each of the first 25 innings. Now, the total runs added in these innings have been contributed by the runs scored in the 26th inning, which must be equal to 25 × 2 = 50 runs.
 
And after contributing 50 runs, his score in the 26th inning is 48 runs.
 
Hence, runs scored in the 26th inning = new average + old innings × change in average
 
= 48 + 25 × 2 = 98
 
To have a mental mapping, we can see the whole situation as:
 
Number of innings
Average in the 1st
25 innings
Average in the 1st
26 innings
Addition
1 46 48 2
2 46 48 2
3 46 48 2
... ... ... ...
... ... ... ...
... ... ... ...
25 46 48 2
26   48  

Properties of Average

  1. Average always lies in between the maximum and the minimum value. It can be equal to the maximum or minimum value if all the numbers are equal.
     
    For example—A1, A2, A3 and A4 are four numbers given where A1 > A2 > A3 > A4.

    Average of these four numbers will always lie in between A1 and A4.
     
    However, if all the four numbers are equal (A1 = A2 = A3 = A4) then the average will be equal to each of these numbers.
     
    Average = A1 = A2 = A3 = A4
  2. Average is the resultant of net surplus and net deficit, as used in the central tendency method.
  3. When weights of different quantities are same, then simple method is used to find the average. However, when different weights of different quantities are taken, then it is known as weighted average. Here the method of weighted average is used to find the average.
     
    For example, assume per capita income of India is USD 500 and per capita income of US is USD 200. Now if we merge India and US into one country then it is observed that per capita income of this new country will not be equal to 2393.png = USD 350
  4. If the value of each quantity is increased or decreased by the same value S, then the average will also increase or decrease respectively by S.
  5. If the value of each quantity is multiplied by the same value S, then the average will also be multiplied by S.
  6. If the value of each quantity is divided by the same value S (S ≠ 0) then the average will also be divided by S.
Example
The average of 4 positive numbers is A and the average of all the possible triples formed out of these four positive numbers is B. Which of the following is true regarding A and B?
  1. A = B
  2. A > B
  3. A < B
  4. Cannot be determined
Solution
Let us assume that the numbers are 1, 2, 3 and 4
 
Average of 1, 2, 3 and 4:1 + 2 + 3 + 4 = 10/4 = 2.5
 
The triplets are 1, 2 and 3; and the average = 2402.png
 
1, 2 and 4; and the average = 2411.png
 
1, 3 and 4; and the average = 2420.png
 
2, 3 and 4; and the average = 2429.png
 
Average of these four averages = 2438.png
 
So, option a (A = B) is the answer.
 

Central value method

It should be observed here that when we find the average of all the possible triplets, all the numbers (1, 2, 3, 4) are added thrice. So effectively we are adding 12 numbers. Hence, the average should be equal to:
 
2447.png

Thus, the average will be equal in all the cases.


Extension to this problem

The average of four positive numbers is A and the average of all the possible pairs formed out of these four positive numbers is B. Which of the following is true?

  1. A = B
  2. A > B
  3. A < B
  4. Cannot be determined




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