Average - Important Points
Simple Average
The term Average refers to the sum of all observations divided by the total number of observations. The term average is also referred to as ‘Mean’. Basic formula to calculate the average is as follows-
Average Formula
Average = Sum of Observations / Number of Observations
Properties of Average
- Average always lies between the least and greatest of given set of observations.
- If each variable in a group increases by k, average also increases by k.
- If each variable is subtracted, multiplied or divided by k, then average will also have the same effect.
Example 1: Four children have 40, 50, 60 and 70 toffees with them. What is the average number of toffees with them?
Solution: Average = (40 + 50 + 60 +70) / 4 = 55
Note: We observe that the average is (40 + 70)/2 = 55 (Mean of middle terms if observations are evenly distributed and number of terms are even. If number of terms is odd, average is the middle term.)
Weighted Average
In weighted average, we emphasise on the concept of “group” instead of “individual” values.
Weighted Average Formula
Average = (Average of Group 1 x Number of Persons + Average of Group 2 x Number of Persons + Average of Group n x Number of Persons) / (n1 + n2 + ....+nn)
Example 2: The average weight of a class of 15 increases by 1 kg, when a student joins the class. Find the weight of the student who joins the class, if the initial average weight of the class is 54 kg.
Solution: Let the weight of the person joining the group be x kg.
Given, (15 × 54 + x)/16 = 55
⇒ x = 16 × 55 – 15 × 54 = 70
Hence the weight of the new person is 70 kg.
Alternate Method: When the new student joins class as the average weight of the class increases by 1 kg, we can understand that this student is bringing 1 kg additionally not only for himself but also for others. Hence his weight should be 54 + 1 (16) i.e., 70 kg
Example 3: The average age of a group of 10 decreased by 1 year when a new person of age 33 joined the group and an existing person left. What is the age of the person who left?
Solution 1: Let the original average age of the group be a and the age of the person who left be y.
⇒ a x 10 + 33 – y = (a – 1) x 10 ⇒ y = 43
Alternate Method: Since the average of 10 person decreased by 1, the sum of the ages of the 10 person should have decreased by 10 years.
So, age of person who left = 10 + 33 = 43