In the previous section we saw Assumed mean method. In this section we will see another method.
We will write it in steps:
1. Consider the table 37.5 that we saw in the previous section. It is shown again below:
• We can see that, all the (di) values are multiples of '15'
♦ In our problem, '15' is the width of the class-intervals
♦ Let us denote this width as 'h'
2. Since they are all multiples of 'h', we can divide each of them by 'h'.
• When we do such a division, the values will become still smaller.
• Those smaller values (denoted as ui) are tabulated in the fifth column in table 37.9 below:
3. Now, we can use (ui) for multiplying with (fi). The products are tabulated in the sixth column
• It is more convenient to calculate (fiui) than (fidi) because ui is smaller than di
4. Once the table is complete, we can calculate u, which is the 'mean of the (ui)s'.
• It is given by the formula:
• Thus for our present problem, we get: u = 29⁄30.
■ Let us compare the formulae for x, d and u:
• While calculating x, we have Σfixi in the numerator
• While calculating d, we have Σfidi in the numerator
• While calculating u, we have Σfiui in the numerator
♦ The denominator is same in all the three cases
• Obviously fidi is easier to calculate than fixi because, di is smaller than xi
♦ Now, fiui is still more easier to calculate than fidi because, ui is smaller than di
4. We have calculated u. But our aim is to find x.
From u, we can easily reach x. The steps are shown below:
• So we can write:
To get x, we simply add the product (hu) to a
• Thus in our problem, x = a + hu = 47.5 + (15 × 29⁄30) = 47.5 + (29⁄2) = 47.5 + 14.5 = 62
• This is the same value we obtained before
■ This method of calculating the mean is called: Step-deviation method
Let us write a summary of what we have seen so far. The summary can be presented in the form of a flow chart:
Now we will see another example:
Example 2:
The table 37.10 below gives the percentage distribution of female teachers in the primary schools of rural areas of various states and union territories (U.T.) of India. Find the mean percentage of female teachers by all the three methods discussed in this section.
Solution:
The three methods are:
(i) Direct method, (ii) Assumed mean method and (iii) Step deviation method
♦ Method (i) has Σfixi in the numerator
♦ Method (ii) has Σfidi in the numerator
♦ Method (iii) has Σfiui in the numerator
• All the three numerators can be conveniently calculated in a single table. It is shown below as table 37.11
• a is taken as 50 and h is taken as 10
1. Direct method:
• x is given by the formula:
• The numerator is calculated at the bottom end of the sixth column. It's value is 1390
• The denominator is calculated at the bottom end of second column. It's value is 35
• So we get x = 1390⁄35 = 39.71
2. Assumed mean method:
• d is given by the formula:
• The numerator is calculated at the bottom end of the seventh column. It's value is -360
• The denominator is calculated at the bottom end of second column. It's value is 35
• So we get d = -360⁄35 = -10.286
• Thus x = a + d = 50 - 10.286 = 39.71
■ This is the same value obtained by method 1
3. Step deviation method:
• u is given by the formula:
• The numerator is calculated at the bottom end of the eighth column. It's value is -36
• The denominator is calculated at the bottom end of second column. It's value is 35
• So we get u = -36⁄35 = -1.0286
• Thus x = a + hu = 50 + (10 × -1.0286) = 50 - 10.286 = 39.71
■ This is the same value obtained by methods 1 & 2
• Though we have obtained the required answers, it is better to do a thorough analysis before proceeding further. This analysis is to find the application of the answer: x = 39.71%
• Consider the table 37.10 which is given to us in the question. From the table, we get many information. We can write the following steps:
1. Consider all the primary schools located in rural areas in a particular state
2. Write the number of female teachers working in all those schools
• Write the number of male teachers working in all those schools
• Write the total number
• Calculate the percentage of female teachers: [Number of female teachers⁄Total number × 100]
3. If the calculated percentage is any value:
♦ equal to or greater than 15
♦ and less than 25
• then that state is included in the first class interval, which is 15 - 25
• The give table shows that there are 6 such states in the country
• In this way, all the class intervals are filled up. The table thus obtained is given to us in the question.
4. We obtained the mean as 39.71
• This is the 'mean of all the percentages'
■ So we can write this in the form of a conclusion:
• Consider all the schools located in rural areas of the whole country
• Consider all the teachers working in those schools
• 39.71 percentage of those teachers is female
• (100-39.71) = 60.29 percentage of those teachers is male
Now we will see some solved examples
Solved example 37.1
A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
Which method did you use to find the mean? why?
Solution:
• We will try all the three methods and then decide which method is appropriate. This will help us to select a method when we encounter such problems in the future. The three methods are:
(i) Direct method, (ii) Assumed mean method and (iii) Step deviation method
♦ Method (i) has Σfixi in the numerator
♦ Method (ii) has Σfidi in the numerator
♦ Method (iii) has Σfiui in the numerator
• All the three numerators can be conveniently calculated in a single table. It is shown below as table 37.13
• a is taken as 7 and h is taken as 2
1. Direct method:
• x is given by the formula:
• The numerator is calculated at the bottom end of the sixth column. It's value is 162
• The denominator is calculated at the bottom end of second column. It's value is 20
• So we get x = 162⁄20 = 8.1
2. Assumed mean method:
• d is given by the formula:
• The numerator is calculated at the bottom end of the seventh column. It's value is 22
• The denominator is calculated at the bottom end of second column. It's value is 20
• So we get d = 22⁄20 = 1.1
• Thus x = a + d = 7 + 1.1 = 8.1
■ This is the same value obtained by method 1
3. Step deviation method:
• u is given by the formula:
• The numerator is calculated at the bottom end of the eighth column. It's value is 11
• The denominator is calculated at the bottom end of second column. It's value is 20
• So we get u = 11⁄20 = 0.55
• Thus x = a + hu = 7 + (2 × 0.55) = 7 + 1.1 = 8.1
■ This is the same value obtained by methods 1 & 2
■ In this problem, the values in the xi column are small. We do not need to simplify them by assuming a mean 'a', or dividing by 'h'. So the Direct method is appropriate.
■ Now we will see the significance of the result: Mean number of plants = 8.1
We will write it in steps:
1. Visit a house and note down the number of plants there
• If the number of plants is zero or 1, that house falls in the first class interval 0-2
• If the number of plants is 2 or 3, that house falls in the second class interval 2-4
so on ...
• In this way the table 37.12 is prepared and is given to us in the question
2. The result we calculated is the 'mean number of plants per house'
• So we can write:
In that locality, there is an average of 8.1 plants per house
3. A similar but simpler example would be:
(i) The following data was obtained from 5 houses in a locality:
• House 1 has 3 plants
• House 2 has 6 plants
• House 3 has 5 plants
• House 4 has 2 plants
• House 5 has 7 plants
(ii) Mean number of plants per house in that locality
= Total number of palnts⁄Total number of houses = (3+6+5+2+7)⁄2 = 23⁄5 = 4.6
• We cannot use this easy method when the data is large
In the next section, we will see a few more solved examples.
We will write it in steps:
1. Consider the table 37.5 that we saw in the previous section. It is shown again below:
 |
| Table.37.5 |
♦ In our problem, '15' is the width of the class-intervals
♦ Let us denote this width as 'h'
2. Since they are all multiples of 'h', we can divide each of them by 'h'.
• When we do such a division, the values will become still smaller.
• Those smaller values (denoted as ui) are tabulated in the fifth column in table 37.9 below:
 |
| Table.37.9 |
• It is more convenient to calculate (fiui) than (fidi) because ui is smaller than di
4. Once the table is complete, we can calculate u, which is the 'mean of the (ui)s'.
• It is given by the formula:
■ Let us compare the formulae for x, d and u:
• While calculating x, we have Σfixi in the numerator
• While calculating d, we have Σfidi in the numerator
• While calculating u, we have Σfiui in the numerator
♦ The denominator is same in all the three cases
• Obviously fidi is easier to calculate than fixi because, di is smaller than xi
♦ Now, fiui is still more easier to calculate than fidi because, ui is smaller than di
4. We have calculated u. But our aim is to find x.
From u, we can easily reach x. The steps are shown below:
• So we can write:
To get x, we simply add the product (hu) to a
• Thus in our problem, x = a + hu = 47.5 + (15 × 29⁄30) = 47.5 + (29⁄2) = 47.5 + 14.5 = 62
• This is the same value we obtained before
■ This method of calculating the mean is called: Step-deviation method
Let us write a summary of what we have seen so far. The summary can be presented in the form of a flow chart:
Example 2:
The table 37.10 below gives the percentage distribution of female teachers in the primary schools of rural areas of various states and union territories (U.T.) of India. Find the mean percentage of female teachers by all the three methods discussed in this section.
 |
| Table.37.10 |
The three methods are:
(i) Direct method, (ii) Assumed mean method and (iii) Step deviation method
♦ Method (i) has Σfixi in the numerator
♦ Method (ii) has Σfidi in the numerator
♦ Method (iii) has Σfiui in the numerator
• All the three numerators can be conveniently calculated in a single table. It is shown below as table 37.11
• a is taken as 50 and h is taken as 10
 |
| Table.37.11 |
• x is given by the formula:
• The denominator is calculated at the bottom end of second column. It's value is 35
• So we get x = 1390⁄35 = 39.71
2. Assumed mean method:
• d is given by the formula:
• The denominator is calculated at the bottom end of second column. It's value is 35
• So we get d = -360⁄35 = -10.286
• Thus x = a + d = 50 - 10.286 = 39.71
■ This is the same value obtained by method 1
3. Step deviation method:
• u is given by the formula:
• The denominator is calculated at the bottom end of second column. It's value is 35
• So we get u = -36⁄35 = -1.0286
• Thus x = a + hu = 50 + (10 × -1.0286) = 50 - 10.286 = 39.71
■ This is the same value obtained by methods 1 & 2
• Though we have obtained the required answers, it is better to do a thorough analysis before proceeding further. This analysis is to find the application of the answer: x = 39.71%
• Consider the table 37.10 which is given to us in the question. From the table, we get many information. We can write the following steps:
1. Consider all the primary schools located in rural areas in a particular state
2. Write the number of female teachers working in all those schools
• Write the number of male teachers working in all those schools
• Write the total number
• Calculate the percentage of female teachers: [Number of female teachers⁄Total number × 100]
3. If the calculated percentage is any value:
♦ equal to or greater than 15
♦ and less than 25
• then that state is included in the first class interval, which is 15 - 25
• The give table shows that there are 6 such states in the country
• In this way, all the class intervals are filled up. The table thus obtained is given to us in the question.
4. We obtained the mean as 39.71
• This is the 'mean of all the percentages'
■ So we can write this in the form of a conclusion:
• Consider all the schools located in rural areas of the whole country
• Consider all the teachers working in those schools
• 39.71 percentage of those teachers is female
• (100-39.71) = 60.29 percentage of those teachers is male
Now we will see some solved examples
Solved example 37.1
A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
 |
| Table.37.12 |
Solution:
• We will try all the three methods and then decide which method is appropriate. This will help us to select a method when we encounter such problems in the future. The three methods are:
(i) Direct method, (ii) Assumed mean method and (iii) Step deviation method
♦ Method (i) has Σfixi in the numerator
♦ Method (ii) has Σfidi in the numerator
♦ Method (iii) has Σfiui in the numerator
• All the three numerators can be conveniently calculated in a single table. It is shown below as table 37.13
• a is taken as 7 and h is taken as 2
 |
| Table.37.13 |
• x is given by the formula:
• The denominator is calculated at the bottom end of second column. It's value is 20
• So we get x = 162⁄20 = 8.1
2. Assumed mean method:
• d is given by the formula:
• The denominator is calculated at the bottom end of second column. It's value is 20
• So we get d = 22⁄20 = 1.1
• Thus x = a + d = 7 + 1.1 = 8.1
■ This is the same value obtained by method 1
3. Step deviation method:
• u is given by the formula:
• The denominator is calculated at the bottom end of second column. It's value is 20
• So we get u = 11⁄20 = 0.55
• Thus x = a + hu = 7 + (2 × 0.55) = 7 + 1.1 = 8.1
■ This is the same value obtained by methods 1 & 2
■ In this problem, the values in the xi column are small. We do not need to simplify them by assuming a mean 'a', or dividing by 'h'. So the Direct method is appropriate.
■ Now we will see the significance of the result: Mean number of plants = 8.1
We will write it in steps:
1. Visit a house and note down the number of plants there
• If the number of plants is zero or 1, that house falls in the first class interval 0-2
• If the number of plants is 2 or 3, that house falls in the second class interval 2-4
so on ...
• In this way the table 37.12 is prepared and is given to us in the question
2. The result we calculated is the 'mean number of plants per house'
• So we can write:
In that locality, there is an average of 8.1 plants per house
3. A similar but simpler example would be:
(i) The following data was obtained from 5 houses in a locality:
• House 1 has 3 plants
• House 2 has 6 plants
• House 3 has 5 plants
• House 4 has 2 plants
• House 5 has 7 plants
(ii) Mean number of plants per house in that locality
= Total number of palnts⁄Total number of houses = (3+6+5+2+7)⁄2 = 23⁄5 = 4.6
• We cannot use this easy method when the data is large
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