Average: Theory & Formulas

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We all know that the average is sum of observations divided by the total number of observations. 
Average Formula = Sum of observations/ Number of observations
This is the simple formula which helps us to calculate the average in math.
Also, let us consider an example, there are five students A, B, C, D, and E having Rs. 5, 10, 15, 20, 25 with them respectively. If we collect all money and divide equally among them, then that equal amount is called the average amount of money.  
Or with the help of formula, average= (5+10+15+20+25)/5=75/5= 15
Let’s see some examples of how we calculate average:
Illustration 1:
Let A, B and C have 20, 25, 15 candies with them respectively. So, what's the average number of candies that they are having with them?
Sol:
Using the formula for calculating average:
Average = Sum of observations/ Number of observations
The sum of observations (candies) is 20 + 25 + 15 = 60
The numbers of observations (persons) = 3
So, answer becomes  60/3 = 20
Illustration 2
Amar, Akbar and Anthony Stole Rs. 3000, Rs. 4000 and Rs. 5000 respectively from different banks. In the evening they sat together and counted how much they had stolen for the day. What was the average amount of money they had stolen from the banks?
Sol:
 Using the formula for calculating average:
Average = Sum of observations/ Number of observations
The sum of observations (Total amount of money) = 3000 + 4000 + 5000 = Rs 12000
The number of observations (Number of men) = 3
Average = 12000/3 = Rs 4000
Some important properties of average are:
When the difference between all the items is same (and the number of terms is odd), then the average is equal to the middle term.
  1. If x is added to all the items, then the average increases by x. 
  2. If x is subtracted from all the items, then the average decreases by x.
  3. If every item is multiplied by x, then the average also gets multiplied by x.
  4. If every item is divided by x, then the average also gets divided by x.
Average formula for An AP Series= (1st term + last term)/2
Illustration 3:
The average number of toffees that the students were given in a school is equal to 25.The principal decided to give the students 25 times the toffees they were given the previous year. What is the average number of toffees the students will get this year?
Sol:
The average number of toffees they had in the previous year was 25.But we know from the above given set of properties that, if every item is multiplied by x, then the average also gets multiplied by x.As the earlier average was 25, so on multiplying the earlier average by 25 as per the given rule the new average becomes 625.
Illustration 4: 
72 students went on a vacation to Ooty. They planned that they will get some amount of money from their parents and their average amount would stand out to be 540.But their parents gave half the amount they had actually planned. What is the new average amount they are taking with them on the vacation?
Sol:
The average amount they were getting earlier was 540 which was reduced to half because of the parents. We know according to the above given properties that, If every item is divided by x, then the average also gets divided by x. As the earlier average was 540, the new average becomes half of the original number, i.e. 540/2 = 270.
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