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To calculate the arithmetic mean using the step-deviation method, follow these steps:

1. **Table Preparation**:
   - The given marks and the corresponding number of students are:
     \[
     \begin{array}{|c|c|}
     \hline
     \text{Marks (X)} & \text{No. of students (f)} \\
     \hline
     5 & 20 \\
     10 & 43 \\
     15 & 75 \\
     20 & 67 \\
     25 & 72 \\
     30 & 45 \\
     35 & 39 \\
     40 & 9 \\
     45 & 8 \\
     50 & 6 \\
     \hline
     \end{array}
     \]
   
2. **Step-Deviation Formula**:
   The formula to find the arithmetic mean using step-deviation is:
   \[
   \bar{X} = A + \left( \frac{\sum f d_i}{\sum f} \right) \cdot h
   \]
   Where:
   - \( A \) is the assumed mean (a suitable value of \( X \)).
   - \( h \) is the class interval (difference between successive values of \( X \)).
   - \( d_i = \frac{X_i - A}{h} \) is the deviation of \( X_i \) from the assumed mean divided by the class interval.
   - \( f \) is the frequency (number of students).

3. **Step-by-Step Calculations**:
   - Assume \( A = 25 \) (since it's the middle value of the marks).
   - Let the class interval \( h = 5 \).

Now, let's proceed to calculate the deviations \( d_i \), their products with frequency \( f d_i \), and finally use the formula to compute the arithmetic mean. I'll calculate that for you.

The arithmetic mean of the marks obtained by the students, calculated using the step-deviation method, is approximately **22.21**.

Would you like further clarification on any part of the process, or additional details?

Here are five related questions to explore:
1. How is the class interval \( h \) determined in step-deviation problems?
2. Why is it helpful to choose an assumed mean near the center of the data?
3. Can the step-deviation method be applied to data without equal intervals?
4. How would the result change if the assumed mean \( A \) were different?
5. How do you interpret the arithmetic mean in this context?

**Tip:** The step-deviation method is useful for simplifying calculations, especially when the data is spread widely across large intervals.

Calculate by step-deviation method, the arithmetic mean of the following marks obtained by students in English.

Learn how to calculate the arithmetic mean of student marks in English using the step-deviation method. This problem includes a detailed solution using core statistical concepts, such as the assumed mean (A), class interval (h), and step-deviation formula. Suitable for students in Grades 10-12.

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