How to calculate the mean using Step deviation method?
The Step Deviation Method is a simplified way to calculate the mean of a grouped frequency distribution, especially when the class intervals are uniform. In simple words, statistics implies the process of gathering, sorting, examine, interpret and then present the data in an understandable manner so as to enable one to form an opinion of it and take necessary action, if necessary. Examples:
- A teacher collecting studentsβ marks, organizing them in ascending or descending manner, and calculating the average class marks, or finding the number of students who failed, informing them so that they start working hard.
- Government officials collecting data for the census, and comparing it with previous records to see whether population growth is in control or not.
- Analyzing the number of followers of a particular religion of a country.
Arithmetic Mean
Arithmetic mean also known as average. Arithmetic Mean for a given set of data is calculated by adding up the numbers in the data and dividing the sum so obtained with the number of observations. It is the most popular method of central tendency.
Properties of Arithmetic Mean
- Deviations from the arithmetic mean of all items in a statistical series would always add up to zero, i.e. β(x β X) = 0.
- The squared deviations from the arithmetic mean is always minimum, i.e., less than the sum of such square deviations from other values like the median, mode, or another tool.
- Replacing all the items in a statistical series with its arithmetic mean has no effect on the sum of the said items.
Using Direct Method
The arithmetic mean is calculated using the following formula,
Sum of observation/Number of observations
Mean of the series = xΜ = Ξ£x/ N.
The formula discussed above pertains to the direct method of calculating the arithmetic mean. But in the case where the calculation becomes tedious owing to larger observations in a data set, other methods can be used to calculate the arithmetic mean, one of such methods being the step-deviation method.
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Step Deviation Method
Whenever the data values are large, and calculation is tedious, the step deviation method is applied.
Steps to find mean using Step Deviation Method
The following steps are used while applying the step deviation method to calculate the arithmetic mean:
- Choose one observation from the data set and mark it as the assumed mean of the whole series. In the case of grouped data, it is not possible to pick an observation from the class intervals, so one first needs to calculate the class marks of mid-points of the intervals and mark one as the assumed mean.
- The next step is to find deviations from the assumed mean (A) by deducting the mean so assumed from all the other observations. d = X β A.
- Next, we are supposed to calculate the step deviations from the deviations obtained above, by finding a common factor, denoted by c, of all the values (deviations), dividing all by this factor, and labelling the step deviations as d1.
- Multiply the step deviations with the frequencies and take up the sum of the numbers so obtained.
- Apply the formula: [Tex]xΜ=a+\frac{Ξ£d_1f}{Ξ£f}Γc[/Tex], where Ξ£d1 is the sum of all the step deviations multiplied by respective frequencies and c represents the common factor.
- The number so obtained is the arithmetic mean of the given data set.
Thus, the formula for the calculation of arithmetic mean by step deviation method is [Tex]xΜ=a+\frac{Ξ£d_1f}{Ξ£f}Γc[/Tex]
Example: Calculate the arithmetic mean for the following data set using the step deviation method:
Marks | Number of Students |
0 β 10 | 5 |
10 β 20 | 12 |
20 β 30 | 14 |
30 β 40 | 10 |
40 β 50 | 5 |
Solution:
Marks
f
m
d = m β A
A = 25
d1 = d/ c
c = 10
fd1
0 β 10
5
5
5 β 25 = β20
β2
β10
10 β 20
12
15
15 β 25 = β10
β1
β12
20 β 30
14
A = 25
25 β 25 = 0
0
0
30 β 40
10
35
35 β 25 = 10
1
10
40 β 50
8
45
45 β 25 = 20
2
16
Ξ£f = 49
Ξ£fd1 = 4
Mean = XΜ = [Tex]a+\frac{Ξ£d_1f}{Ξ£f}Γc[/Tex]
= [Tex]25+\frac{4}{49}Γ10 [/Tex]
= 25 + 0.81
= 25.81
Hence, Arithmetic Mean of the given data set is 25.81
Sample questions on calculating the mean using Step deviation method
Question 1. Calculate the mean using the step deviation method:
Marks | Number of students |
10 β 20 | 5 |
20 β 30 | 3 |
30 β 40 | 4 |
40 β 50 | 7 |
50 β 60 | 2 |
60 β 70 | 6 |
70 β 80 | 13 |
Solution:
Marks f
m
d = m β A
A = 45
d1 = d/ c
c = 10
fd1
10 β 20 5
15 β30
β3
β15
20 β 30 3
25 β20
β2
β6
30 β 40 4
35 β10
β1
β4
40 β 50 7
45 0
0
0
50 β 60 2
55 10
1
2
60 β 70 6
65 20
2
12
70 β 80 13
75 30
3
39
Ξ£f = 40
Ξ£fd1 = 28 Mean = XΜ = [Tex]a+\frac{Ξ£d_1f}{Ξ£f}Γc[/Tex]
= [Tex]45+\frac{28}{40}Γ10[/Tex]
= 45 + 7
= 52
Hence, Arithmetic Mean of the given data set is 52.
Question 2. Calculate the mean using the step deviation method:
Class Intervals | Frequency |
β40 to β30 | 10 |
β30 to β20 | 28 |
β20 to β10 | 30 |
β10 to 0 | 42 |
0 to 10 | 65 |
10 to 20 | 180 |
20 to 30 | 10 |
Solution:
Class Intervals
f
m
d = m β A
A = β5
d1 = d/c
c = 10
fd1
β40 to β30
10
β35
β30
β3
β30
β30 to β20
28
β25
β20
β2
β56
β20 to β10
30
β15
β10
β1
β30
β10 to 0
42
β5
0
0
0
0 to 10
65
5
10
1
65
10 to 20
180
15
20
2
180
20 to 30
10
25
30
3
30
Ξ£f = 365 Ξ£fd1 = 159 Mean = XΜ = [Tex]a+\frac{Ξ£d_1f}{Ξ£f}Γc[/Tex]
= [Tex]β5+\frac{159}{365}Γ10[/Tex]
= β0.64
Hence arithmetic mean is β0.64
Question 3. Calculate the mean using the step deviation method:
Wages | Number of workers |
0 β 10 | 22 |
10 β 20 | 38 |
20 β 30 | 46 |
30 β 40 | 35 |
40 β 50 | 19 |
Solution:
Wages
f
m
d = m β A
A = 25
d1 = d/c
c = 10
fd1
0 β 10
22
5
β20
β2
β44
10 β 20
38
15
β10
β1
β38
20 β 30
46
25
0
0
0
30 β 40
35
35
10
1
35
40 β 50
19
45
20
2
38
Ξ£f = 160
Ξ£fd1 = β9
Mean = XΜ = [Tex]a+\frac{Ξ£d_1f}{Ξ£f}Γc[/Tex]
= [Tex]25+\frac{β9}{160}Γ10[/Tex]
= 24.44
Hence the arithmetic mean is 24.44
Question 4. Calculate the mean using the step deviation method:
Age | Number of People |
0 β 20 | 4 |
20 β 40 | 10 |
40 β 60 | 15 |
60 β 80 | 20 |
80 β 100 | 11 |
Solution:
Age
f
m
d = m β A
A = 50
d1 = d/c
c = 20
fd1
0 β 20
4
10
β40
β2
β8
20 β 40
10
30
β20
β1
β10
40 β 60
15
50
0
0
0
60 β 80
20
70
20
1
20
80 β 100
11
90
40
2
22
Ξ£f = 60 Ξ£fd1 = 24 Mean = XΜ = [Tex]a+\frac{Ξ£d_1f}{Ξ£f}Γc[/Tex]
= [Tex]50+\frac{24}{60}Γ20[/Tex]
= 50 + 8
= 58
Hence arithmetic mean is 58.
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