How Do We Calculate the Mean of Grouped Data by the Step Deviation Method?

The step deviation method is a simplified way to calculate the mean of grouped data by reducing the size of the deviations, making calculations easier. Here's how it works:

Steps to Calculate the Mean Using the Step Deviation Method:

1. Create a frequency distribution table: This table should include class intervals and their corresponding frequencies (f_i).

2. Find the midpoint (x_i) of each class interval: The midpoint is calculated as (Upper Class Limit + Lower Class Limit) / 2.

3. Assume a mean (A): Choose a midpoint (x_i) that is roughly in the middle of the data. This assumed mean simplifies calculations.

4. Calculate the class size (h): This is the difference between the upper and lower limits of a class interval, assuming that the class size is constant.

5. Calculate the step deviations (u_i): This is the key step in simplifying the calculation. Use the following formula:

   u_i = (x_i - A) / h

6. Calculate f_iu_i for each class interval: Multiply the frequency (f_i) of each class interval by its corresponding step deviation (u_i).

7. Calculate the sum of f_iu_i (Σf_iu_i) and the sum of frequencies (Σf_i): Add up all the values from the f_iu_i column and the frequency column.

8. Apply the Step Deviation Formula:

   Mean (x̄) = A + (h * (Σf_iu_i / Σf_i))

   Where:

   • x̄ is the mean.
   • A is the assumed mean.
   • h is the class size.
   • Σf_iu_i is the sum of the product of frequencies and step deviations.
   • Σf_i is the sum of frequencies.

Example:

Let's say we have the following grouped data:

Class Interval	Frequency (fi)
10-20	5
20-30	8
30-40	12
40-50	7
50-60	3

1. Midpoints (x_i): 15, 25, 35, 45, 55
2. Assume Mean (A): Let's assume A = 35
3. Class Size (h): h = 10
4. Step Deviations (u_i): (15-35)/10 = -2, (25-35)/10 = -1, (35-35)/10 = 0, (45-35)/10 = 1, (55-35)/10 = 2
5. f_iu_i: 5*(-2) = -10, 8*(-1) = -8, 12*0 = 0, 7*1 = 7, 3*2 = 6
6. Σf_iu_i = -5, Σf_i = 35
7. *Mean (x̄) = 35 + (10 (-5/35)) = 35 - (50/35) = 35 - 1.43 = 33.57**

Therefore, the mean of the grouped data using the step deviation method is approximately 33.57.

Benefits of the Step Deviation Method:

• Simplifies calculations: By reducing the size of the deviations, the method avoids dealing with large numbers, especially when the midpoints and assumed mean are significantly large.
• Reduces errors: Smaller numbers lead to fewer calculation errors.

In summary, the step deviation method is an efficient way to calculate the mean of grouped data, offering a more manageable approach compared to other methods, especially when dealing with large datasets and class sizes.