Step Deviation Method
Step deviation method is a method of obtaining the mean of grouped data when the values are large. In statistics, there are three types of mean  arithmetic mean, geometric mean, and harmonic mean. When the values of the data are large and the deviation of the class marks have common factors, the step deviation method is used. Let us learn more about the formula, the derivation of the step deviation method formula, and solve a few examples.
1.  Definition of Step Deviation Method 
2.  Step Deviation Method Formula 
3.  Derivation of Step Deviation Method 
4.  Steps for Using Step Deviation Method 
5.  FAQs on Step Deviation Method 
Definition of Step Deviation Method
Step deviation method can be defined as the method used to obtain the mean of large values which is divisible by a common factor. These values of deviations are reduced to a smaller value by dividing all the values by a common factor. In other words, the step deviation method is used when the deviations of the class marks from the assumed mean are large and they all have a common factor. This method is also called a change of origin or scale method. The mean of grouped data is obtained by three different methods, direct method, assumed or shortcut method, and step deviation method. The step deviation method is considered as the extension of the assumed method as we use the deviation formula used in the assumed method.
Step Deviation Method Formula
Before we go to the step deviation formula, let us quickly look at both the direct method and assumed or shortcut method formulas.
Estimated or Direct Mean = ∑x\(_i\)f\(_i\) / ∑f\(_i\), where f\(_i\) is the frequency and x\(_i\) is the midpoint of the class interval.
Assumed Mean = A + ∑d\(_i\)f\(_i\) / ∑f\(_i\), where A is the assumed mean, f\(_i\) is the frequency, and deviation d\(_i\) = x\(_i\)  A.
Since this method is the extension of the assumed mean method, the formula is:
Step Deviation of Mean = A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)],
where,
 A is the assumed mean
 h is the class size
 u\(_i\) = d\(_i\)/h
 f\(_i\) is the frequency
 d\(_i\) = x\(_i\)  A
 x\(_i\) is the midpoint of the class interval
Derivation of Step Deviation Method
Let us derive the step deviation formula by using the direct method formula and the deviation process in the assumed mean method. Let us consider the class size to be h and the assumed mean to be A. Using the same formula used in assumed mean, d\(_i\) = x\(_i\)  A and calculating the value of u\(_i\) with the equation u\(_i\) = d\(_i\)/h, where h is the class width and d\(_i\) = x\(_i\)  A. Hence, we can derive the formula as:
x̄ = ∑d\(_i\)f\(_i\) / ∑f\(_i\)
x̄ = ∑[u\(_i\)h + A]f\(_i\) / ∑f\(_i\)
x̄ = ∑[u\(_i\)f\(_i\)h + Af\(_i\)] / ∑f\(_i\)
x̄ = ∑u\(_i\)f\(_i\)h + ∑Af\(_i\) / ∑f\(_i\)
x̄ = [∑u\(_i\)f\(_i\)h] / ∑f\(_i\) + ∑Af\(_i\) / ∑f\(_i\)
x̄ = h[∑u\(_i\)f\(_i\)] / ∑f\(_i\) + A∑f\(_i\) / ∑f\(_i\)
x̄ = A + h[∑u\(_i\)f\(_i\)] / ∑f\(_i\)
Steps for Using Step Deviation Method
Steps to be followed while applying the step deviation method are given below,
 Create a table containing five columns as given below,
Column 1 Class interval.
Column 2 Classmarks (corresponding), denoted by x\(_i\). Take the central value from the class marks as the Assumed Mean and denote it as A.
Column 3 Calculate the corresponding deviations given as, i.e. di = x\(_i\)  A
Column 4 Calculate the values of u\(_i\) using the formula, u\(_i\) = d\(_i\)/h, where h is the class width.
Column 5 Frequencies (f_{i}) (corresponding)  Find the Mean of u\(_i\) = ∑x\(_i\)u\(_i\) / ∑u\(_i\)
 Finally, calculate the Mean by adding the assumed mean A to the product of class width(h) with mean of u\(_i\)
Let us look at an example to understand this better.
Example: Consider the following example to understand this method. Find the mean of the following using the stepdeviation method.
Class Intervals  010  1020  2030  3040  4050  5060  6070  Total 
Frequency  4  4  7  10  12  8  5  50 
Solution: To find the mean, we first have to find the class marks and decide A (assumed mean). Let A = 35 Here h (class width) = 10
C.I.  x_{i}  f_{i}  u_{i}= xi−A/h  f_{i}u_{i} 
010  5  4  3  4 x (3)=12 
1020  15  4  2  4 x (2)=8 
2030  25  7  1  7 x (1)=7 
3040  35  10  0  10 x 0= 0 
4050  45  12  1  12 x 1=12 
5060  55  8  2  8 x 2=16 
6070  65  5  3  5 x 3=15 
Total  ∑fi=50  ∑fiui=16 
Using mean formula:
x̄ = A + h × ∑x\(_i\)u\(_i\) / ∑u\(_i\) = 35 + (16/50) ×10 = 35 + 3.2 = 38.2
Mean = 38.
Related Topics
Listed below are a few topics related to the step deviation method, take a look.
Examples on Step Deviation Method

Example 1: Find the mean of the following data.
Class Interval 50  70 70  90 90  110 110  130 130  150 150  170 Frequency 15 10 20 22 16 17 Solution: The first step is to create the table with the midpoint or marks and the product of the frequency and midpoint. To calculate the midpoint we find the average between the class intervals.
Midpoint x\(_i\) = 50  70 = 60 ([70 + 50]/2), 70  90 = 80 ([90 + 70]/2) and so on. From the midpoint let us select the assumed mean, so A = 100 and the value of h = 20 which is the class size.
Find the value, u\(_i\) = d\(_i\)/h, where d\(_i\) = x\(_i\)  A. Hence, we can write it as u\(_i\) = x\(_i\)  A / h. Let us find the value for each class interval. So, 50  70 = (60  100) / 20 = 2 , 70  90 = (80  100) / 20 = 1 and so on.
f\(_i\) × d\(_i\) =  20 × 12 =  240 ,  10 × 15 =  150 and so on.
Class Interval (f(_i\)) (x\(_i\)) u\(_i\) = x\(_i\)  A / h f\(_i\)u\(_i\) 50  70 15 60  2  30 70  90 10 80  1  10 90  110 20 100 0 0 110  130 22 120 1 22 130  150 16 140 2 32 150  170 17 160 3 51 Total 100 65 Once we have determined the totals, let us use the assume mean formula to calculate the estimated mean.
Step Deviation of Mean = A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)]
= 100 + 20 [65/100]
= 100 + 13
= 113.
Therefore, the mean of the data is 113.

Example 2: Find the mean percentage of the work completed for a project in a country where the assumed mean is 50, the class size is 20, frequency is 100, and the product of the frequency and deviation is  42. Solve this by using the stepdeviation method.
Solution: Given,
a = 50, h = 20, f\(_i\) = 100, f\(_i\)u\(_i\) =  42
Using the step deviation method formula,
Step Deviation of Mean = a + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)]
= 50 + 20 [42/100]
= 50  42/5
= 50  8.4
= 41.6
Therefore, the mean percentage is 41.6.
FAQs on Step Deviation Method
What is Meant by Step Deviation Method?
Step deviation method is the extended method of the assumed or shortcut method of obtaining the mean of large values. These values of deviations are divisible by a common factor that is reduced to a smaller value. The step deviation method is also called a change of origin or scale method.
What is the Formula for Step Deviation Method?
To obtain the mean of large values, the formula of the step deviation method is applied. It is as follows:
Step Deviation of Mean = A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)],
where,
 A is the assumed mean
 h is the class size
 u\(_i\) = d\(_i\)/h
 f\(_i\) is the frequency
 d\(_i\) = x\(_i\)  A
 x\(_i\) is the midpoint of the class interval
When Should We Use Step Deviation Method?
To obtain the mean of data, there are three methods that can be used direct method, assumed method, and step deviation method. The direct method helps in obtaining the mean when the values are small, the assumed method helps in obtaining the mean when the values are in between, and the step deviation method is used for obtaining the mean when large values.
How Do You Calculate Step Deviation Method?
The following steps are used to calculate the mean using this method:
 Calculate the class marks of each class x\(_i\)
 Let A denote the assumed mean of the data.
 Find u\(_i\) =(x\(_i\)−A)/h, where h is the class size.
 Use the formula: A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)]
What is the Common Factor in Step Deviation Method?
The common factor is the greatest positive number that divides the deviations. For instance, if deviations are –20, –10, 0, 40, and 60, then the common factor is 10 because 10 is the greatest positive number that divides the five numbers.
What is Step Deviation Method for Calculating Arithmetic Mean?
Arithmetic mean for grouped data can be calculated by using step deviation method when the values are large. The formula is:
Step Deviation of Mean = A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)],
where,
 A is the assumed mean
 h is the class size
 u\(_i\) = d\(_i\)/h
 f\(_i\) is the frequency
 d\(_i\) = x\(_i\)  A
 x\(_i\) is the midpoint of the class interval
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