How to Calculate Mean using Step Deviation Method?

Whenever the data values are large, and calculation is tedious, the step deviation method is applied.

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.
• If the class intervals are not continuous, then one must first make them continuous by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval.
• 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: , 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 

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

Solution: Class intervals are Continuous

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 = 

= 

= 25 + 0.81

= 25.81

Hence, Arithmetic Mean of the given data set is 25.81

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

Solution: Class intervals are not Continuous

The given class intervals do not touch each other.
The gap between successive intervals is 1.

To make them continuous, subtract 0.5 from every lower limit and add 0.5 to every upper limit.

After that, we apply the step-deviation method.

Class Interval	Continuous Interval	f	m	d = m - A A = 37	d1 = d/ c c = 3	fd1
30 - 32	29.5–32.5	5	31	31 - 37 = −6	−2	−10
33 - 35	32.5–35.5	12	34	34 - 37 = −3	−1	−12
36 - 38	35.5–38.5	18	A = 37	37 - 37 = 0	0	0
39 - 41	38.5–41.5	7	40	40 - 37 = 3	1	7
42 - 44	41.5–44.5	8	43	43 - 37 = 6	2	16
	Σf = 50	Σfd1 =1

Mean = X̄ = 

= 

= 37 + 0.06

= 37.06

Hence, Mean of the given data set is 37.06

Sample Questions on Calculating 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 = 

= 

= 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	360
20 to 30	10	25	30	3	30
Σf = 365	Σfd1 = 339

Mean = X̄ = 

= 

= 4.288

Hence arithmetic mean is 4.288

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 = 

= 

= 24.44

Hence, 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 = 

= 

= 50 + 8

= 58

Hence, arithmetic mean is 58.

Practice Problems: Step Deviation Method

Calculate the mean using the step deviation method.

Problem 1: Consider the following frequency distribution:

Problem 2: Consider the following frequency distribution:

Problem 3: Consider the following frequency distribution:

Problem 4: Consider the following frequency distribution: