Step Deviation Method in Statistics – Definition, Formula, and Solved Example

Updated: August 13, 2025

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Category: Advance Math, Math

The Step Deviation Method is a technique used in statistics to simplify the calculation of the mean for grouped data, particularly when dealing with large class intervals. It’s an extension of the assumed mean method, further simplifying calculations by reducing the deviations to smaller, more manageable values.

Key Concepts and Steps

Class Marks (xᵢ): First, determine the midpoint of each class interval (xᵢ). This is done by adding the upper and lower limits of the interval and dividing by two.
Assumed Mean (a): Choose an assumed mean (a) from the class marks. This is typically a value near the center of the distribution.
Deviations (dᵢ): Calculate the deviation (dᵢ) of each class mark from the assumed mean: dᵢ = xᵢ – a.
Step Deviations (uᵢ): Divide each deviation (dᵢ) by the class size (h) to obtain the step deviations (uᵢ): $u_i = \frac{d_i}{h}$
Frequency Multiplication: Multiply each step deviation (uᵢ) by its corresponding frequency (fᵢ) to get fᵢuᵢ.
Summation: Calculate the sum of fᵢuᵢ (∑fᵢuᵢ) and the sum of frequencies (∑fᵢ).
Mean Calculation: Apply the formula:
$\text{Mean} = a + h \times \frac{\sum f_i u_i}{\sum f_i}$

When to Use the Step Deviation Method

The step deviation method is particularly useful when:

• The data is grouped into class intervals.
• The class intervals are uniform (have the same width).
• The values are large, making direct calculation of the mean cumbersome.

Formula of Step Deviation Method

Mean (x̄) = a + h × (∑fᵢuᵢ / ∑fᵢ)

Where:
a = Assumed mean
h = Class interval width
fᵢ = Frequency of each class
uᵢ = Step deviation = (xᵢ – a) / h

Step-by-Step Procedure

Prepare the table with class intervals and frequencies.
Find midpoints (xxx) of each class.
Select an assumed mean (A).
Calculate deviations: d=x−A
Find step deviations: u=d/h
Multiply f × u times for each class.
Apply the formula to get the mean.

Example – How to Calculate Mean using Step Deviation Method

Imagine a dataset of student marks with class intervals like 10–20, 20–30, 30–40, and so on. If you have a large number of students, calculating the mean using the direct method could be tedious. The step deviation method simplifies this by reducing the deviations to smaller, more manageable numbers, making the calculation faster and easier.

Example of Step Deviation Method

Class Interval	Frequency (f)
0–10	5
10–20	8
20–30	10
30–40	7
40–50	5

Step 1: Class width (h) = 10
Step 2: Midpoints (xᵢ): 5, 15, 25, 35, 45
Step 3: Assume a = 25
Step 4: Deviations (dᵢ = xᵢ – a): -20, -10, 0, 10, 20
Step 5: Step deviations (uᵢ = dᵢ / h): -2, -1, 0, 1, 2
Step 6: fᵢuᵢ: -10, -8, 0, 7, 10
Step 7: ∑fᵢ = 35, ∑fᵢuᵢ = -1
Step 8: Mean = 25 + 10 × (-1 / 35) = 25 – 0.2857 ≈ 24.71

Applications of Step Deviation Method

• Used in educational statistics to calculate average marks in exams.
• Helpful in economic data analysis for simplifying large datasets.
• Common in industrial statistics for production and quality control.

Conclusion

The step deviation method is an efficient statistical technique for finding the mean of grouped data with large values and uniform intervals. By converting data into smaller step deviations, it reduces calculation complexity and saves time. This makes it a preferred method in academic, economic, and industrial data analysis.

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