What is a step deviation method explained with an example?

The step deviation method is a simplified approach to calculating the mean of grouped data. It involves reducing the computational complexity by assuming a new variable to simplify the arithmetic calculations. This method is particularly useful when the class intervals and frequencies are large, making calculations cumbersome.

### Steps of Step Deviation Method:
1. **Choose an Assumed Mean (A):** Select one of the class midpoints as the assumed mean (A). This reduces the size of numbers during calculations.
2. **Calculate Class Midpoints (xi):** For each class interval, compute the midpoint using the formula:
xi = (Lower Limit + Upper Limit) / 2
3. **Find Deviations (di):** Calculate the deviation of each midpoint from the assumed mean using the formula:
di = xi - A
4. **Divide Deviations by Class Width (ui):** Simplify the deviations by dividing each di by the class width (h):
ui = di / h
5. **Multiply Frequencies (fi) with ui (fi * ui):** Calculate the product of frequencies (fi) and the reduced deviation (ui) for each class.
6. **Find Σfi and Σ(fi * ui):** Compute the total frequency (Σfi) and the sum of the products (Σ(fi * ui)).
7. **Apply the Formula for Mean (X̄):** Use the formula to find the mean:
X̄ = A + (Σ(fi * ui) / Σfi) * h

### Example:
Let us calculate the mean for the following grouped data:

| Class Interval | Frequency (fi) |
|----------------|-----------------|
| 0 - 10 | 5 |
| 10 - 20 | 8 |
| 20 - 30 | 15 |
| 30 - 40 | 16 |
| 40 - 50 | 6 |

1. **Calculate Class Midpoints (xi):**
Midpoints = (0+10)/2, (10+20)/2, (20+30)/2, (30+40)/2, (40+50)/2
xi = 5, 15, 25, 35, 45

2. **Choose Assumed Mean (A):** Assume A = 25 (from the midpoint of the class 20-30).

3. **Calculate Deviations (di):**
di = xi - A = 5 - 25, 15 - 25, 25 - 25, 35 - 25, 45 - 25
di = -20, -10, 0, 10, 20

4. **Divide by Class Width (h = 10):**
ui = di / h = -20/10, -10/10, 0/10, 10/10, 20/10
ui = -2, -1, 0, 1, 2

5. **Multiply Frequencies with ui:**
fi * ui = 5*(-2), 8*(-1), 15*0, 16*1, 6*2
fi * ui = -10, -8, 0, 16, 12

6. **Compute Σfi and Σ(fi * ui):**
Σfi = 5 + 8 + 15 + 16 + 6 = 50
Σ(fi * ui) = -10 - 8 + 0 + 16 + 12 = 10

7. **Apply the Formula:**
X̄ = A + (Σ(fi * ui) / Σfi) * h
X̄ = 25 + (10 / 50) * 10
X̄ = 25 + 2
X̄ = 27

### Final Answer:
The mean of the grouped data is 27. This step deviation method simplifies the computation while maintaining accuracy.