Averages
The averages or mean of a set of numbers is that which represents the central value of the whole set of numbers.
MEASURES OF AVERAGES
 Mode
 Median
 Arithmetic Mean (AM)
 Geometric Mean (GM)
Remember
If in a given set of numbers, two or more values occur the same number of times, then an unique mode does not exist.
MODE
Mode, means the most frequent value assumed by a random variable i.e. the number that occurs most frequently in a given set of numbers.
Solved Examples (Mode)
 Find the mode of 1, 2, 3, 3, 3, 4, 4, 4, 1, 2, 3, 5, 1, 1, 1.
a) 3
b) 5
c) 4
d) 1
Sol. Â Number of 1’s = 5
2’s = 2
3’s = 4
4’s = 3
5â€™s = 1
As the frequency of 1 is maximum, the mode is 1.
Hence, option d.
 Find the mode of 20, 21, 21, 20, 22, 22, 23, 44, 45, 20, 21, 22.
a) 21
b) 20
c) 23
d) Cannot be determined
Sol.Â Number of 20â€™s = 3
21’s = 3
22â€™s = 3
23’s = 1
44â€™s = 1
45â€™s = 1
As the frequency of 20, 21 and 22 is the same, we cannot determine a unique mode.
Hence, option (d).
MEDIAN
Arranging the terms in ascending or descending order reveals the median as the middle term.
Solved Examples (Median)
 Find the median of 1, 3, 5, 7, 9, 11, 13, 15, 17.
a) 5
b) 7
c) 9
d) Cannot be determined
 Find the median of 2, 14, 16, 8, 10, 12, 4, 6.
a) 6
b) 8
c) 9
d) 10
ARITHMETIC MEAN (AM)

Types of Arithmetic Mean (AM)

Simple Arithmetic Mean (AM)
Important concept to note
 If the value of each item is increased or decreased by the same value p, then the average of the group or items will also increase or decrease by p.
 If the value of each item is multiplied or divided by the same value p, then the average of the group or items will also be multiplied or divided by p.
 The average of a group of items will always lie between the smallest value in the group and the largest value in the group i.e., the average will be greater than the smallest value and less than the largest value in the group.
Solved Examples (Arithmetic Mean)
 Find the arithmetic mean of 1, 3, 5, 7, 9, 11, 13, 15, 17,
a) 5
b) 7
c) 9
d) Cannot be determined
 If in a final examination consisting of 8 papers Anna scored 91, 92, 98, 85, 94, 95, 85, 96 respectively, find her average score.
a) 92
b) 94
c) 93.5
d) 91.5
Geometric Mean (GM)
Solved Examples (Geometric Mean)
 Find the Geometric Mean of 128, 64, 32, 16, 8, 4, 2.
a) 16
b) 8
c) 6
d) 4
 There are two groups A and B respectively. There are 8 persons in group A and 9 persons in group B. If the geometric mean of group A is 3 and that of group B is 2 then find the geometric mean of the combined group.
a) 416
b) 456
c) 615
d) 728
Concepts of Averages
1st ConceptÂ
2nd ConceptÂ
3rd ConceptÂ
4th ConceptÂ
5th ConceptÂ
When the average of a group changes as a result of one element being replaced by another; the value of the element coming in can be found in the following way:
Value of the element coming in = Value of element being replaced + (increase in average Ã— no. of elements)
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