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 – 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]x̄=a+\frac{Σd_1f}{Σf}×c[/Tex], where Σd₁ 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]x̄=a+\frac{Σd_1f}{Σf}×c[/Tex]

Example: Calculate the arithmetic mean for the following data set using the step deviation method:

Solution:

Marks	f	m	d = m – A A = 25	d1 = d/ c c = 10	fd1
0 – 10	5	5	5 – 25 = −20	−2	−10
10 – 20	12	15	15 – 25 = −10	−1	−12
20 – 30	14	A = 25	25 – 25 = 0	0	0
30 – 40	10	35	35 – 25 = 10	1	10
40 – 50	8	45	45 – 25 = 20	2	16
	Σf = 49				Σfd1 = 4

Mean = X̄ = [Tex]a+\frac{Σd_1f}{Σf}×c[/Tex]

= [Tex]25+\frac{4}{49}×10 [/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:

Solution:

Marks	f	m	d = m – A A = 45	d1 = d/ c c = 10	fd1
10 – 20	5	15	−30	−3	−15
20 – 30	3	25	−20	−2	−6
30 – 40	4	35	−10	−1	−4
40 – 50	7	45	0	0	0
50 – 60	2	55	10	1	2
60 – 70	6	65	20	2	12
70 – 80	13	75	30	3	39
	Σf = 40				Σfd1 = 28

Mean = X̄ = [Tex]a+\frac{Σd_1f}{Σf}×c[/Tex]

= [Tex]45+\frac{28}{40}×10[/Tex]

= 45 + 7

= 52

Hence, Arithmetic Mean of the given data set is 52.

Question 2. Calculate the mean using the step deviation method:

Solution:

Class Intervals	f	m	d = m – A A = −5	d1 = d/c c = 10	fd1
−40 to −30	10	−35	−30	−3	−30
−30 to −20	28	−25	−20	−2	−56
−20 to −10	30	−15	−10	−1	−30
−10 to 0	42	−5	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
	Σf = 365				Σfd1 = 159

Mean = X̄ = [Tex]a+\frac{Σd_1f}{Σf}×c[/Tex]

= [Tex]−5+\frac{159}{365}×10[/Tex]

= −0.64

Hence arithmetic mean is −0.64

Question 3. Calculate the mean using the step deviation method:

Solution:

Wages	f	m	d = m – A A = 25	d1 = d/c c = 10	fd1
0 – 10	22	5	−20	−2	−44
10 – 20	38	15	−10	−1	−38
20 – 30	46	25	0	0	0
30 – 40	35	35	10	1	35
40 – 50	19	45	20	2	38
	Σf = 160				Σfd1 = −9

Mean = X̄ = [Tex]a+\frac{Σd_1f}{Σf}×c[/Tex]

= [Tex]25+\frac{−9}{160}×10[/Tex]

= 24.44

Hence the arithmetic mean is 24.44 

Question 4. Calculate the mean using the step deviation method:

Solution:

Age	f	m	d = m – A A = 50	d1 = d/c c = 20	fd1
0 – 20	4	10	−40	−2	−8
20 – 40	10	30	−20	−1	−10
40 – 60	15	50	0	0	0
60 – 80	20	70	20	1	20
80 – 100	11	90	40	2	22
	Σf = 60				Σfd1 =  24

Mean = X̄ = [Tex]a+\frac{Σd_1f}{Σf}×c[/Tex]

= [Tex]50+\frac{24}{60}×20[/Tex]

= 50 + 8

= 58

Hence arithmetic mean is 58.

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 students’ 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.