How do you calculate the standard deviation from the following data by step deviation method?

To calculate the standard deviation using the step deviation method, you need data. Assuming we are dealing with grouped data, the following steps outline the process and formula used. If you are working with ungrouped data, a simplified version of these steps applies. Crucially, I need the data to provide a complete calculation. This response provides the method assuming you can supply the data.

Understanding the Step Deviation Method

The step deviation method simplifies standard deviation calculations, especially when dealing with large numerical values or grouped data. It involves creating a simplified scale using a common factor and assumed mean.

Steps for Calculating Standard Deviation (Grouped Data) via Step Deviation

1. Organize the Data: Create a frequency distribution table with class intervals (if the data is continuous) or distinct values (if the data is discrete). Let's say we have n class intervals.

2. Find the Midpoint (x_i): For each class interval, determine the midpoint (class mark). This is calculated as (Upper Class Limit + Lower Class Limit) / 2. If you have discrete data, the 'x_i' values are your data points.

3. Assume a Mean (A): Choose a convenient value from the midpoints (x_i) as the assumed mean (A). Ideally, choose a midpoint near the center of the data to simplify calculations.

4. Calculate the Step Deviation (d'_i): Calculate the step deviation for each class using the formula:

   d'_i = (x_i - A) / h

   where:

   • x_i is the midpoint of the i^th class interval
   • A is the assumed mean
   • h is the class width (the common factor or the width of the class interval, assuming equal class sizes.)

5. Calculate f_id'_i: Multiply the frequency (f_i) of each class by its corresponding step deviation (d'_i).

6. Calculate f_i(d'_i)²: Multiply the frequency (f_i) of each class by the square of its step deviation (d'_i)².

7. Calculate ∑f_id'_i and ∑f_i(d'_i)²: Sum the values obtained in steps 5 and 6.

8. Apply the Formula: Calculate the standard deviation (σ) using the following formula:

   σ = h * √[ (∑f_i(d'_i)² / N) - (∑f_id'_i / N)² ]

   where:

   • h is the class width (step size)
   • N is the total frequency (∑f_i)
   • ∑f_i(d'_i)² is the sum of the product of frequency and the square of the step deviation
   • ∑f_id'_i is the sum of the product of frequency and the step deviation

Example (Illustrative, needs your data for actual values):

Let's say you provide the following grouped data (this is just an example structure):

Class Interval	Frequency (fi)
0-10	5
10-20	8
20-30	12
30-40	7
40-50	3

You would then follow the steps above to complete the calculation. The assumed mean (A) could be, for example, 25 (the midpoint of the 20-30 interval). The class width (h) is 10.

Ungrouped Data

For ungrouped data, a similar but simplified approach is used. The formula simplifies to:

σ = h * √[ (∑(d')² / n) - (∑d'/ n)² ]

Where d' = (x - A) / h, 'x' represents each data point, 'A' is an assumed mean, 'h' is a common factor, and 'n' is the number of data points.

Conclusion

The step deviation method offers a streamlined approach to calculate standard deviation, particularly useful with grouped or large datasets. By utilizing an assumed mean and a common factor, the calculations become more manageable and less prone to errors. Provide the relevant data for a complete and accurate calculation.