Step Deviation Method: Formula, Steps & Examples

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The step deviation method refers to determining the mean of grouped data when the values are large. In statistics, there are three kinds of mean: arithmetic mean, geometric mean, and harmonic mean.

The step deviation method is utilised when the deviation of the class marks has common factors.
It is also known as a shift of origin and scale method.
The step deviation method is an extended version of the assumed mean method used to calculate the mean of large values.
In this method, a discrete data set is used for calculation.
The method has a fixed interval between the observations.

Table of Content

Step Deviation Method
Step Deviation Method Formula
Derivation of Step Deviation Method
Steps for Using Step Deviation Method
Things to Remember
Sample Questions

Key Terms: Step Deviation Method, Statistics, Data, Mean, Mean of Grouped Data, Assumed Mean Method, Step Deviation Method Formula, Derivation of Step Deviation Method

Step Deviation Method

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The step deviation method calculates the mean of big numbers that can be divisible by a relation of common factors. By dividing all of the variables with a common factor, the deviations are simplified to a smaller number.

In other words, when the deviations of the class marks from the expected mean are considerable and they all consist of common factors, the step deviation approach is applied.
Three approaches are used to calculate the mean of grouped data: direct method, assumed or short-cut method, and step deviation method.
Since we apply the deviation formula, the step deviation method is considered an extension of the assumed approach.

Step Deviation Method

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Step Deviation Method Formula

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Let's take a quick look at the direct method and assumed method formulas before moving on to the step deviation formula.

Estimated or Direct Mean = ∑x_if_i/ ∑f_i, here f_i refers to the frequency and x_i refers to the CenterPoint of the class interval.

Assumed Mean = A + ∑d_if_i / ∑f_i, here, A refers to the assumed mean, f_i refers to frequency, and deviation d_i = x_i- A.

Step Deviation of Mean is the extension of the assumed mean method.

Step Deviation of Mean = A + h [∑u_if_i / ∑f_i]

Here,

A refers to the assumed mean,
h refers to class size,
u_i= d_i/h,
f_irefers to frequency,
d_i = x_i- A, and,
x_irefers to the centre point of the class interval.

Example of Step Deviation Method Formula

Example: Calculate the mean percentage of completed work of a project where the assumed mean is 100, the class size is 20, the frequency is 100, and the product of the frequency and variation is - 42. Solve it with the help of step-deviation method?

Ans: Given, a = 100, h = 20, f_i = 100, f_iu_i= - 42

Step Deviation of Mean = A + h [∑u_if_i / ∑f_i]

= 100 + 20 [-42/100]

= 100 - 42/5

= 100 - 8.4

= 91.6

Hence, the mean percentage is 91.6.

Derivation of Step Deviation Method

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We can use the direct method formula and the deviation process in the assumed mean method to derive the standard deviation formula. So, assume that the class size is h and the assumed mean is A.

With the help of same formula as in assumed mean, d_i = x_i- A and calculating the value of ui as u_i= d_i/h where h is the class width.
As a result, we can derive the formula as follows:

⇒ x̄ = ∑d_if_i/ ∑f_i

⇒ x̄ = ∑[u_ih + A]f_i / ∑f_i

⇒ x̄ = ∑[u_if_ih + Af_i] / ∑f_i

⇒ x̄ = ∑u_if_ih + ∑Af_i / ∑f_i

⇒ x̄ =[∑u_if_ih] / ∑f_i + ∑Af_i / ∑f_i

⇒ x̄ = h[∑u_if_i] / ∑f_i + ∑Af_i / ∑f_i

∴ x̄ = A + h[∑u_if_i] / ∑f_i

Steps for Using Step Deviation Method

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The steps to be taken when using the step deviation method are listed below:

Make a table with five columns, starting with Column 1 which indicates the class interval.
Column 2 indicates the class marks (corresponding), represented by x_i.
As the Assumed mean , use the central value from the class marks and denote it as A.
Column 3 is used to determine the corresponding deviations, i.e. d_i = x_i- A
Column 4 is used to determine the u_ivalues using the formula u_i = d_i/h, where h is the class width.
Column 5 represents the corresponding frequencies (f_i).
Determine the mean of ui = ∑x_iu_i / ∑u_i
Finally, compute the Mean by multiplying the assumed mean A by the product of class width(h) and the mean of u_i.

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Things to Remember

The step deviation approach is used to get the mean of grouped data.
With the help of it we can calculate the mean of big numbers which are divisible by a common factor.
It is the shortcut of assumed method.
The step deviation method is used to determine accurate values by hand.
This method is used when there is a common component and significant differences between the class marks and the expected mean.

Sample Questions

Ques: The marks scored by 5 students in a weekly class test are 7, 9, 6, 4, 2 out of 10. Determine the mean marks of the class? (2 marks)

Ans: Formula to find mean marks of the class are:

[Average marks = Sum of observation / Number Of Observation]

Here average marks = (7 + 9 + 6 + 4 + 2) / 5 = 28 / 5 = 5.6

Therefore, 5.6 is the mean mark for the class.

Ques: The mean and variance of 7 observations are 8 and 16 respectively. If five of the observations are 2,4,10,12 and 14. Now determine the remaining two observations? (5 marks)

Ans: Let’s take the remaining two observation as x and y.
So the observation are 2,4,10,12,14,x,y
Mean =x?=72+4+10+12+14+x+y=8
⇒56=42+x+y
⇒x+y=14 ......(1)

Variance =16=n1i=l∑7(Xi−X?)2
16=71[(−6)2+(−4)2+(2)2+(4)2+(6)2+x2+y2−2×8(x+y)+2×(8)2]
16=71[36+16+4+6+36+x2+y2−16(14)+2(64)]   ...........[using (1)]
16=71[108+x2+y2−224+128]
16=71[12+x2+y2]
⇒x2+y2=112−12=100
x2+y2=100   ..........(2)
From (1), we obtain
x2+y2+2xy=196   ....(3)
From (2) and (3), we obtain
2xy=196−100
⇒2xy=96 ...........(4)
subtracting (4) from (2), we obtain
x2+y2−2xy=100−96
⇒(x−y)2=4
⇒x−y=±2   ............(5)
Therefore, from (1) and (5) we obtain
x=8 and y=6 when x−y=2
x=6 and y=8 when x−y=−2

Hence, the remaining observation are 6 and 8.

Ques: Calculate the mean percentage of completed work of a project where the assumed mean is 50, the class size is 20, the frequency is 100, and the product of the frequency and variation is - 42. Solve it with the help of step-deviation method? (2 marks)

Ans: Given, a = 50, h = 20, fi = 100, fiui = - 42

Step Deviation of Mean = a + h [∑uifi / ∑fi]

= 50 + 20 [-42/100]

= 50 - 42/5

= 50 - 8.4

= 41.6

Hence, the mean percentage is 41.6.

Ques: Determine the mean of the following given info set. 10, 20, 36, 12, 35, 40, 36, 30, 36, 40? (3 marks)

Ans: Given, xi = 10, 20, 36, 12, 35, 40, 36, 30, 36, 40

n = 10

Mean = ∑xi/n

= (10 + 20 + 36 + 12 + 35 + 40 + 36 + 30 + 36 + 40)/10

= 295/10

= 29.5

Hence, the mean of the given data set is 29.5.

Ques: The average weight of the nine students is 24 kg. If one more student is added to the group, the mean remains unchanged, so the weight of the tenth student is? (2 marks)

Ans: The sum of weights of the nine students = 25×9=22525×9=225 kg

If one more student adds up in the group, then the total number of students is 10, and the mean is 25.

The sum of weights of the 10 students is 25×10=25025×10=250 kg

The weight of the tenth student is 250−225=25250−225=25 kg

Ques: The arithmetic mean is a measure of central tendency and is commonly referred to as the mean. The arithmetic mean is calculated by dividing the total of all the values in a series by the number of items in that series. Arithmetic mean is normally denoted by X¯X¯ which is red as 'XX bar.' It can be computed for unclassified or ungrouped data, as well as discrete or continuous series, as well as classified or grouped data. Calculate the arithmetic mean from the following data? (4 marks)

Marks	0-10	10-20	20-30	30-40	40-50	50-60
No. of students	10	20	30	50	40	30

Ans: Let us take assumed mean = 45
Calculation of deviations from assumed mean

Marks (X)	m	No. of students (f)	*(X-45)/10* (d)	fd
0-10	5	10	-4	-40
10-20	15	20	-3	-60
20-30	25	30	-2	-60
30-40	35	50	-2	-50
40-50	45	40	0	0
50-60	55	30	+1	30

N ==180

∑fd=∑fd=-180
Mean = A+∑fdN×c=45+−180×10180=35

Ques: The table given below shows frequency distribution of advertisement on TV by shift of origin and scale method. Determine mean duration of advertisement on T.V. by shift of origin and scale method? (4 marks)

Duration (in sec.)	25-30	30-35	35-40	40-45	45-50	50-55
No. of advertisements	10	32	15	9	7	2

Ans: Consider the following table, to calculate mean by "shift of origin method":

xixi=mid value of class interval

Assumed mean a=42.5a=42.5

cici	fifi	xixi	di=xi−adi=xi−a	fidifidi
25−3025−30	1010	27.527.5	−15−15	−150−150
30−3530−35	3232	32.532.5	−10−10	−320−320
35−4035−40	1515	37.537.5	−5−5	−75−75
40−4540−45	99	42.542.5	00	00
45−5045−50	77	47.547.5	55	3535
50−5550−55	22	52.552.5	1010	2020

N=Σfi=75N=Σfi=75

Σfidi=−490Σfidi=−490

Mean x¯=a+ΣfidiNx¯=a+ΣfidiN

∴x¯=42.5+−49075=35.966667⇒35.97∴x¯=42.5+−49075=35.966667⇒35.97

Hence, Mean duration of advertisement on T.V is 35.9735.97 seconds

Ques: The marks scored by 5 students in a weekly class test are 70, 9, 16, 24, 2 out of 10. Determine the mean marks of the class? (2 marks)

Ans: Formula to find mean marks of the class are:

[Average marks = Sum of observation / Number Of Observation]

Here average marks = (70 + 9 + 16 + 24 + 2) / 5 = 121 / 5 = 24.2

Therefore, 24.2 is the mean mark for the class.

Ques: Calculate the mean using the step deviation method? (4 marks)

Age	Number of People
0 – 20	40
20 – 40	10
40 – 60	50
60 – 80	20
80 – 100	10

Ans: The process is as follows:

Age	f	m	d = m – A and A = 50	d1 = d/c c = 20	fd1
0 – 20	40	10	−40	−2	−80
20 – 40	10	30	−20	−1	−10
40 – 60	50	50	0	0	0
60 – 80	20	70	20	1	20
80 – 100	10	90	40	2	20
	Σf = 130				Σfd1 = -50

Mean = 50 + (-50/130) x 20

Hence arithmetic mean is -42.3.

Ques: Calculate the mean percentage of completed work of a project where the assumed mean is 150, the class size is 20, the frequency is 100, and the product of the frequency and variation is - 40. Solve it with the help of step-deviation method? (2 marks)

Ans: Given, a = 150, h = 20, fi = 100, fiui = - 40

Step Deviation of Mean = a + h [∑uifi / ∑fi]

= 150 + 20 [-40/100]

= 150 - 8

= 142

Hence, the mean percentage is 142.

Ques: Calculate the mean percentage of completed work of a project where the assumed mean is 50, the class size is 20, the frequency is 200, and the product of the frequency and variation is - 20. Solve it with the help of step-deviation method? (2 marks)

Ans: Given, a = 50, h = 20, fi = 200, fiui = - 20

Step Deviation of Mean = a + h [∑uifi / ∑fi]

= 50 + 20 [-20/100]

= 50 - 4

= 46

Hence, the mean percentage is 46.

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Step Deviation Method: Formula, Steps & Examples

Step Deviation Method

Step Deviation Method Formula

Example of Step Deviation Method Formula

Derivation of Step Deviation Method

Steps for Using Step Deviation Method

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

2.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

3.
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Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

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At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

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Step Deviation Method: Formula, Steps & Examples

Step Deviation Method

Step Deviation Method Formula

Example of Step Deviation Method Formula

Derivation of Step Deviation Method

Steps for Using Step Deviation Method

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

2. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

3.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

Comments

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1.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

2.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

3.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :