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: \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.