To calculate the mean using the step-deviation method, we'll follow these steps:

Step 1: Determine the class intervals, midpoints, and frequencies

Class Interval	Midpoint ( $x_i$ )	Frequency ( $f_i$ )
25 - 35	$\frac{25+35}{2} = 30$	6
35 - 45	$\frac{35+45}{2} = 40$	10
45 - 55	$\frac{45+55}{2} = 50$	8
55 - 65	$\frac{55+65}{2} = 60$	12
65 - 75	$\frac{65+75}{2} = 70$	4

Step 2: Choose an assumed mean ( $A$ )

We can choose any midpoint as the assumed mean. Let's take $A = 50$.

Step 3: Calculate the deviation ( $d_i = x_i - A$ )

Midpoint ( $x_i$ )	$d_i = x_i - 50$
30	$30 - 50 = -20$
40	$40 - 50 = -10$
50	$50 - 50 = 0$
60	$60 - 50 = 10$
70	$70 - 50 = 20$

Step 4: Calculate the step-deviation ( $u_i = \frac{d_i}{h}$ )

We need to determine the class width, $h$, which is the difference between two consecutive midpoints. Here, $h = 10$.

Deviation ( $d_i$ )	$u_i = \frac{d_i}{h}$
-20	$\frac{-20}{10} = -2$
-10	$\frac{-10}{10} = -1$
0	$\frac{0}{10} = 0$
10	$\frac{10}{10} = 1$
20	$\frac{20}{10} = 2$

Step 5: Calculate $f_i \times u_i$

Frequency ( $f_i$ )	Step-deviation ( $u_i$ )	$f_i \times u_i$
6	-2	$6 \times (-2) = -12$
10	-1	$10 \times (-1) = -10$
8	0	$8 \times 0 = 0$
12	1	$12 \times 1 = 12$
4	2	$4 \times 2 = 8$

Step 6: Use the formula for the mean

The formula for the mean using the step-deviation method is: $\bar{x} = A + h \times \frac{\sum f_i u_i}{\sum f_i}$

Now, let's calculate $\sum f_i u_i$ and $\sum f_i$:

$\sum f_i u_i = -12 + (-10) + 0 + 12 + 8 = -2$ $\sum f_i = 6 + 10 + 8 + 12 + 4 = 40$

Now, substitute the values into the formula:

$\bar{x} = 50 + 10 \times \frac{-2}{40}$ $\bar{x} = 50 + 10 \times (-0.05)$ $\bar{x} = 50 - 0.5 = 49.5$

Final Answer:

The mean marks using the step-deviation method is 49.5.

Would you like more details or have any other questions? Here are some related queries:

1. What is the purpose of choosing an assumed mean in the step-deviation method?
2. How do we decide the class width $h$ in grouped data?
3. Can you explain the difference between the step-deviation method and the direct method for finding the mean?
4. How do we compute the variance using grouped data?
5. Why is the midpoint important in calculating the mean for grouped data?

Tip: Choosing an assumed mean close to the actual mean can simplify calculations.