In the previous section we saw the details about Mean. In this section we will see some solved examples.
Solved example 25.19
The table (a) below shows the students in a class sorted according to the marks they got for a test.
Calculate the mean mark of the class.
Solution:
1. We have:
2. Table (b) is prepared by expanding table (a)
• The numerator is calculated at the bottom end of third column in table (b). It's value is 297
• The denominator is calculated at the bottom end of second column in table (b). It's value is 45
3. So we get x = 297/45 = 6.6
Solved example 25.20
The table below shows the days in a month sorted according to the amount of rainfall in a locality.
What is the mean rainfall per day during this month?
Solution:
1. We have:
2. Table below is prepared by expanding the given table
• The numerator in (1) is calculated at the bottom end of third column in the above table. It's value is 1545
• The denominator is calculated at the bottom end of second column in the above table. It's value is 30
3. So we get x = 1545/30 = 51.5
Solved example 25.21
The details of rubber sheets that a farmer got during a month are given below:
(i) How many kilograms of rubber did he get a day on average in this month?
(ii) The price of rubber is Rs. 120 per kg. What is his average income per day this month from selling rubber?
Solution:
1. We have:
2. Table below is prepared by expanding the given table
• The numerator in (1) is calculated at the bottom end of third column in the above table. It's value is 381
• The denominator is calculated at the bottom end of second column in the above table. It's value is 30
3. So we get x = 381/30 = 12.7
Now we will see a peculiarity of mean:
1. Consider any 8 numbers between 100 and 200, both inclusive
What is the least possible sum of those 8 numbers?
Ans: The least possible sum will be obtained when all those 8 numbers are equal to the least possible value 100. This will give the sum 800
2. What is the mean of those 8 numbers?
Ans: Mean = 800/8 = 100
This is the least possible mean
3. What is the maximum possible sum of those 8 numbers?
Ans: The maximum possible sum will be obtained when all those 8 numbers are equal to the maximum possible value 200. This will give the sum 1600
4. What is the mean of those 8 numbers?
Ans: Mean = 1600/8 = 200
This is the maximum possible mean
■ The least possible mean is 100, and the maximum possible mean is 200. So, if the numbers are any other than 100 or 200, the mean will lie in between 100 and 200, both inclusive.
■ So we can write this:
• There are 8 numbers which lie in between 100 and 200, both numbers inclusive.
• The mean of those 8 numbers will also lie between 100 and 200, both inclusive
This is true for any set of numbers. We can write it in the form of a theorem:
Theorem 25.1
The mean of any set of numbers between two fixed numbers is also between those two numbers
Ans: Then also, the least possible sum is when all the numbers are 100.
That sum = 100 × (sum of all frequencies)
2. So least possible Mean = [100 × (sum of all frequencies)]⁄(sum of all frequencies) = 100
3. Also, the maximum possible sum is when all the numbers are 200.
That sum = 200 × (sum of all frequencies)
4. So maximum possible Mean = [200 × (sum of all frequencies)]⁄(sum of all frequencies) = 200
■ So we find that, theorem 25.1 is valid in this case also
In the next section we will see Median and Mode
Solved example 25.19
The table (a) below shows the students in a class sorted according to the marks they got for a test.
Solution:
1. We have:
• The numerator is calculated at the bottom end of third column in table (b). It's value is 297
• The denominator is calculated at the bottom end of second column in table (b). It's value is 45
3. So we get x = 297/45 = 6.6
Solved example 25.20
The table below shows the days in a month sorted according to the amount of rainfall in a locality.
Solution:
1. We have:
• The denominator is calculated at the bottom end of second column in the above table. It's value is 30
3. So we get x = 1545/30 = 51.5
Solved example 25.21
The details of rubber sheets that a farmer got during a month are given below:
(ii) The price of rubber is Rs. 120 per kg. What is his average income per day this month from selling rubber?
Solution:
1. We have:
• The denominator is calculated at the bottom end of second column in the above table. It's value is 30
3. So we get x = 381/30 = 12.7
1. Consider any 8 numbers between 100 and 200, both inclusive
What is the least possible sum of those 8 numbers?
Ans: The least possible sum will be obtained when all those 8 numbers are equal to the least possible value 100. This will give the sum 800
2. What is the mean of those 8 numbers?
Ans: Mean = 800/8 = 100
This is the least possible mean
3. What is the maximum possible sum of those 8 numbers?
Ans: The maximum possible sum will be obtained when all those 8 numbers are equal to the maximum possible value 200. This will give the sum 1600
4. What is the mean of those 8 numbers?
Ans: Mean = 1600/8 = 200
This is the maximum possible mean
■ The least possible mean is 100, and the maximum possible mean is 200. So, if the numbers are any other than 100 or 200, the mean will lie in between 100 and 200, both inclusive.
■ So we can write this:
• There are 8 numbers which lie in between 100 and 200, both numbers inclusive.
• The mean of those 8 numbers will also lie between 100 and 200, both inclusive
Theorem 25.1
The mean of any set of numbers between two fixed numbers is also between those two numbers
1. In the above example, what if 'frequency' is involved?
That is., what if some of the numbers occur more than once. Ans: Then also, the least possible sum is when all the numbers are 100.
That sum = 100 × (sum of all frequencies)
2. So least possible Mean = [100 × (sum of all frequencies)]⁄(sum of all frequencies) = 100
3. Also, the maximum possible sum is when all the numbers are 200.
That sum = 200 × (sum of all frequencies)
4. So maximum possible Mean = [200 × (sum of all frequencies)]⁄(sum of all frequencies) = 200
■ So we find that, theorem 25.1 is valid in this case also
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