Average

Average:
Average is commonly known as average. The average of a given set of numbers is called the Average, or simply, the mean of the given numbers. So , the Average of a group of observations is defined as

Mean = Sum of observations / Number of observations

x is the symbol of the Average.
So , the mean of n observation x1, x2, . . ., xn, is given by

Properties of Average:
Property 1: If x is the Average of n observations x1, x2, x3, . . xn; then (x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x) = 0.
Property 2: The mean of n observations x1, x2, x3, . . xn is x. If each observation is increased by p, the mean of the new observations is (x + p).
Property 3: The mean of n observations x1, x2, x3, . . xn is x. If each observation is decreased by p, the mean of the new observations is (x - p).
Property 4: The mean of n observations x1, x2, x3, . . xn is x. If each observation is multiplied by a nonzero number p, the mean of the new observations is px.
Property 5: The mean of n observations x1, x2, x3, . . xn is x. If each observation is divided by a nonzero x number p, the mean of the new observations is x/p

Problems based on average:
1. The heights of five runners are 164 cm, 137 cm, 149 cm, 149 cm and 161 cm respectively. Find the mean height per runner.
Solution:
Mean height = Sum of the heights of the runners/number of runners
(164 + 137 + 149 + 149 + 161 ) / 5
= 760 / 5
= 152 cm.
So, the mean height is 152 cm.
>2. . Find the mean of the first six prime numbers.
Solution:
The first six prime numbers are 2, 3, 5, 7 ,11 and13.

Mean = Sum of the first six prime numbers / number of prime numbers
(2 + 3 + 5 + 7 + 11+13 ) / 6
= 41/6
= 6.833

So, their mean is 6.833

3. Find the mean of the first six multiples of 4.
Solution:
The first six multiples of 4 are 4, 8, 12, 16, 20 and 24.

Mean = Sum of the first six multiples of 4 / number of multiples
(4 + 8 + 12 + 16 + 20 + 24) / 6
= 84 / 6
= 14.
So, their mean is 14.

4. Find the Average of the first 8 natural numbers.
Solution:
The first 8 natural numbers are 1, 2, 3, 4, 5, 6 , and 7 and 8.
Let x denote their Average.

Then mean = Sum of the first 7 natural numbers/number of natural numbers (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 )
x = 36 / 8
= 4.5
So, their mean is 4.5

5. If the mean of 9, 8, 10, x, 12 is 15, find the value of x.
Solution:
Mean of the given numbers = (9 + 8 + 10 + x + 12)/5 = (39 + x)/5
According to the problem, mean = 15 (given).
So, (39 + x)/5 = 15
? 39 + x = 15 x 5
? 39 + x = 75
? 39 + x = 75 - 39
? x = 36
So, x = 36.

6. The mean of 40 numbers was found to be 38. Later on, it was detected that a number 56 was misread as 36. Find the correct mean of given numbers.
Solution:
Calculated mean of 40 numbers = 38.
So, calculated sum of these numbers = (38 x 40) = 1520.
Correct sum of these numbers
= [1520 - (wrong item) + (correct item)]
= (1520 - 36 + 56)
= 1540.
So, the correct mean = 1540/40 = 38.5.

7. The mean of the heights of 6 males is 152 cm. If the individual heights of five of them are 151 cm, 153 cm, 155 cm, 149 cm and 154 cm, find the height of the sixth boy.
Solution:
Mean height of 6 males = 152 cm.
Sum of the heights of 6 males = (152 x 6) = 912 cm
Sum of the heights of 5 males = (151 + 153 + 155 + 149 + 154) cm = 762 cm.
Height of the sixth boy
= (sum of the heights of 6 males) - (sum of the heights of 5 males)
= (912 - 762) cm = 150 cm.
So, the height of the sixth female is 150 cm.

8. The mean weight of a group of seven males is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.
Solution:
Mean weight of 7 males = 56 kg.
Total weight of 7 males = (56 x 7) kg = 392 kg.
Total weight of 6 males = (52 + 57 + 55 + 60 + 59 + 55) kg
= 338 kg.
Weight of the 7th boy = (total weight of 7 males) - (total weight of 6 males)
= (392 - 338) kg
= 54 kg.
So, the weight of the seventh boy is 54 kg.

9. A cricketer has a mean score of 58 runs in nine innings. Find out how many runs are to be scored by him in the tenth innings to raise the mean score to 61.
Solution:
Mean score of 9 innings = 58 runs.
Total score of 9 innings = (58 x 9) runs = 522 runs.
Required mean score of 10 innings = 61 runs.
Required total score of 10 innings = (61 x 10) runs = 610 runs.
Number of runs to be scored in the 10th innings
= (total score of 10 innings) - (total score of 9 innings)
= (610 -522) = 88.
So, the number of runs to be scored in the 10th innings = 88.

10. The average height of 30 males was calculated to be 150 cm. It was detected later that one value of 165 cm was wrongly copied as 135 cm for the computation of the mean. Find the correct mean.
Solution:
Calculated average height of 30 males = 150 cm.
Incorrect sum of the heights of 30 males
= (150 x 30)cm
= 4500 cm.
Correct sum of the heights of 30 males
= (incorrect sum) - (wrongly copied item) + (actual item)
= (4500 - 135 + 165) cm
= 4530 cm.
Correct mean = correct sum/number of males
= (4530/30) cm
= 151 cm.
So, the correct mean height is 151 cm.

11. The mean of 16 items was found to be 30. On rechecking, it was found that two items were wrongly taken as 22 and 18 instead of 32 and 28 respectively. Find the correct mean.
Solution:
Calculated mean of 16 items = 30.
Incorrect sum of these 16 items = (30 x 16) = 480.
Correct sum of these 16 items
= (incorrect sum) - (sum of incorrect items) + (sum of actual items)
= [480 - (22 + 18) + (32 + 28)]
= 500.
So, correct mean = 500/16 = 31.25.
So, the correct mean is 31.25.