Step Deviation Method

The step deviation method is a statistical technique that is used to calculate the mean of a grouped data efficiently and easily. It simplifies the calculation by selecting an assumed mean, determining the class midpoints and class width to standardize deviations. This method reduces the larger numbers into manageable values. This makes it useful for datasets with uniform class intervals.
 

The formula for step deviation is given below:


Mean =  \( A + c \frac{\Sigma f_i u_i}{\Sigma f_i}\)
 
Where,

c is the class width

A is the assumed mean

Σf_iu_i is the sum of the product of frequency and deviation values

Σf_i is the number of frequencies

Find the mean of the following frequency distribution using the step deviation method.

Class Interval: 10–20, 20–30, 30–40, 40–50, 50–60.

Frequency (f): 5, 8, 15, 16, 6

Applying the step deviation method,

The assumed mean of the data is \(A = 35.\)

The step value, or class width, of the data is \(h = 10.\)

Let us now apply the step deviation method.

\(\bar{x} = A + \left( \frac{\sum f_i u_i}{\sum f_i} \right) h\)

\(\sum f_i = 5 + 8 + 15 + 16 + 6 = 50 \)

\(\sum f_i u_i = -10 - 8 + 0 + 16 + 12 = 10\)

So,

\(\bar{X}=35+(\frac{10}{50})×10\\[1em] \bar{X}=35+(0.2×10)\\[1em] \bar{X}=35+2=37\)

Steps to be followed while applying the step deviation method are given below,

Step 1: Create a table containing five columns.

Step 2: Name the first column as the class interval.

Step 3: Name the second column as class marks, denoted by xi. Take the central value from this section as the assumed mean and denote it as A. 

Step 4: In the third column, calculate the corresponding deviations. Calculate it using the formula, di=xi-A. 

Step 5: Calculate the values of ui using the formula, ui = di/h, where h is the class width, in the fourth column. 

Step 6: Write the frequencies in the fifth column. 

Step 7: Find the mean of ui with the formula, ui = ∑xiui / ∑ui

Step 8: Now, we can calculate the mean by adding the assumed mean A to the product of the class width and the mean of ui.

Step deviation method is very helpful when we are collecting a data that is very large and when it speaks about many different things. Here are some tips and tricks that would help learners of all age in mastering step deviation method. 

• Teachers can start teaching the step deviation method by explaining when to use it. Before getting into the complex formulas, teach them why we are using them. We can use it when the intervals are equal and when the numbers are large.
   
• Parents can tell their children that step deviation is like converting big rocks into small pebbles. We use the step deviation method to save time. 
   
• Learners often get stuck while calculating the midpoints. Show them the easy shortcut to calculate the midpoint. 
  
  \(\text{Midpoint = Lower limit} + \frac{h}{2}\)
   
• Parents can encourage their children to use a multicolored table to easily identify the midpoints, the assumed mean, deviations, and \(f-u_i\) values. 
   
• Encourage learners to estimate before calculating. Ask them, “What would be the mean?” This builds their number sense and awareness of errors. 
   
• Students should try to explain the steps once they practice. This helps a lot in building their confidence and mastery over the subject.

The step deviation method has numerous applications across various fields. Let us explore how the step deviation method is used in different areas:
 

• Education: The step deviation method is used by teachers and researchers to calculate the average marks of students from large data sets. It also helps in analyzing the distribution of student performance in different subjects.
  
   
• Economics: The step deviation method helps economists to determine the average income of individuals in different income brackets. It is also used to study income inequality by analyzing grouped data of households' income.
  
   
• Business: Step deviation method is used by companies to calculate the average sales revenue for different product categories. It also helps in understanding customer purchase behavior.