“Average is a very simple but effective way
of representing an entire group by a single value. .
“Average” of a group is defined as
“Sum of all the items in the group” means “sum
of the values of all the items in the group”. .
Example, Let us say a cricketer played 9
innings in a year. Let us say he scored the following runs in those innings 35,
56, 124, 29, 0, 87, 98, 45 and 75. Then his average score (per innings) for the
year is .
Total score/Number of innings = = 61
Similarly the average height of a class of
students is equal to the sum of the heights of all the students of the class
divided by the number of students in the class. .
Average is also called the “mean” or mean
value of all the values. .
Points to Remember
- If the value of each item is increased by the
same value p, then the average of the group or items will also increase by p. .
- If the value of each item is decreased by the
same value p, then the average of the group or items will also decrease by p. .
- If the value of each item is multiplied by
the same value p, then the average of the group or items will also he
multiplied by p. .
- If the value of each item is divided by the
same value p (p > 0), then the
average of the group or items will also be divided by p. .
- The average of a group of distinct items will
always lie between the smallest value in the group and 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. .
An Easy Method to Calculate Averages
As already discussed, the average of a group
of items whose values are given can be found out by the rule given at the
beginning of this section. However, in most of the cases, we do not need to
perform such elaborate additions and divisions. The calculation of averages can
be simplified greatly by taking some arbitrary number (P) as a starting point,
take the deviations of the given items (Q1) from this arbitrary
number, find the average of all these deviations (Q1 - P) and add it
to the arbitrary number (P) to give the correct average of the given items. .
If there are n items and they are denoted by
Q1, Q2, Q3, ……Qn, then the average
of these n items is given by
Average = P +
The extent to which this method will simplify
the calculation will depend on the selection of the arbitrary value P. It
should be selected in such a way that the positive and negative deviations
cancel out each other to the extent possible. Then the final figure left for
division will be relatively small making the division easier. .
For example, the cricketer that we considered
above scored the following runs in seven innings: 35, 56, 45, 43, 67, 70 and
48. Now, to find his average, we take an arbitrary figure, say 50 and first
find the deviations of each of the scores from this figure. The deviations of
the scores from 50 are -15, +6, -5, -7, +17, +20 and -2. The sum of these
deviations is +14. .
Hence the average of the cricketer’s scores
50 + = 52
Please note that the number P (= 50 above)
can be any value. Let us work out the same example taking a different value for
P. Let us take P equal to 45. The deviations of the scores from P are -10, + 11,
0, -2, +22, +25 and +3. The sum of these deviations is 49. Hence the average is
45 + 49/7 = 45 + 7 = 52.
When two groups of items are combined
together, then we can talk of the average of the entire group. However, if we
know only the average of the two groups individually, we cannot find out the
average of the combined group of items. .
For example, there are two sections A and B
of a class where the average height of section A is 150 cm and that of section
B is 160 cm. On the basis of this information alone, we cannot find the average
of the entire class (of the two sections). As discussed earlier, the average
height of the entire class is .
Since we do not have any information
regarding the number of students in the two sections, we cannot find the
average of the entire class. Now, suppose that we are given that there are 60
students in the section A and 40 students in the section B, then we can
calculate the average height of the entire class which, in this case will be
equal to = 154 cm.
This average height 154 cm of the entire
class is called “weighted average” of the class. .
The above step in calculating the weighted
average of the class can be rewritten as below:
3/5 and 2/5 are the weights multiplying the
averages. Hence the average height of the entire class is called as weighted
Some solved example
The monthly incomes of Ajay in January,
February and March this year are Rs.3000, Rs.4000 and Rs.5000 respectively.
Find his average monthly income for these three months. .
= = Rs.4000
Three numbers have an average of 20. If two
of the numbers are 14 and 28, the third number is .
the three numbers = (20) (3) = 60
Third number = 60 - (sum of the other two
numbers) = 60 - 42 = 18
Vijay purchased 1 dozen mangoes at Rs.6 per
dozen, 2 dozen mangoes of another variety at Rs.10 per dozen and 5 dozen
mangoes of a third variety at Rs.6 per dozen. Find the average cost per dozen
of mangoes purchased by Vijay.
Cost of 1 dozen mangoes = (1) (6) = Rs.6.
Cost of 2 dozen mangoes = (2) (10) = Rs.20 .
Cost of 5 dozen mangoes = (5) (6) = Rs.30.
Total cost of 8 dozen mangoes = Rs.56 .
Average cost per dozen of mangoes = = Rs.7
The average weight of a group of 4 girls is
25 kg. A girl joins them and the average weight of the group goes up by 1 kg.
Find the weight of the girl who joined. .
weight of the 4 girls = (4) (25)
= 100 kg
Total weight of 5 girls after the girl joins
them = (5) (26) = 130 kg
Weight of the girl who joined = 130 – 100
= 30 kg