Calculate the mean and standard deviation using step deviation method for the following data :
`{:("Class marks"(x_(i)),15,30,45,60,75,90,105,120),("Frequency" (f_(i)),12,14,65,107,157,202,222,230):}`

Answer

To calculate the mean and standard deviation using the step deviation method for the given data, we will follow these steps:

### Step 1: Organize the Data
We have the following class marks (x_i) and their corresponding frequencies (f_i):

| Class Marks (x_i) | Frequency (f_i) |
|-------------------|-----------------|
| 15 | 12 |
| 30 | 14 |
| 45 | 65 |
| 60 | 107 |
| 75 | 157 |
| 90 | 202 |
| 105 | 222 |
| 120 | 230 |

### Step 2: Choose an Assumed Mean (A)
Let's choose the assumed mean (A) as 75 (a value from the class marks).

### Step 3: Calculate Deviations (D_i)
Calculate the deviations (D_i) from the assumed mean (A):

\[ D_i = x_i - A \]

Calculating D_i for each class mark:

- For x_1 = 15: D_1 = 15 - 75 = -60
- For x_2 = 30: D_2 = 30 - 75 = -45
- For x_3 = 45: D_3 = 45 - 75 = -30
- For x_4 = 60: D_4 = 60 - 75 = -15
- For x_5 = 75: D_5 = 75 - 75 = 0
- For x_6 = 90: D_6 = 90 - 75 = 15
- For x_7 = 105: D_7 = 105 - 75 = 30
- For x_8 = 120: D_8 = 120 - 75 = 45

### Step 4: Calculate Step Deviations (u_i)
The class size (h) is the difference between two consecutive class marks. Here, h = 15.

Calculate step deviations (u_i):

\[ u_i = \frac{D_i}{h} \]

Calculating u_i:

- u_1 = -60 / 15 = -4
- u_2 = -45 / 15 = -3
- u_3 = -30 / 15 = -2
- u_4 = -15 / 15 = -1
- u_5 = 0 / 15 = 0
- u_6 = 15 / 15 = 1
- u_7 = 30 / 15 = 2
- u_8 = 45 / 15 = 3

### Step 5: Calculate f_i * u_i
Now, we will calculate the product of frequency and step deviation (f_i * u_i):

| Class Marks (x_i) | Frequency (f_i) | u_i | f_i * u_i |
|-------------------|-----------------|------|-----------|
| 15 | 12 | -4 | -48 |
| 30 | 14 | -3 | -42 |
| 45 | 65 | -2 | -130 |
| 60 | 107 | -1 | -107 |
| 75 | 157 | 0 | 0 |
| 90 | 202 | 1 | 202 |
| 105 | 222 | 2 | 444 |
| 120 | 230 | 3 | 690 |

### Step 6: Calculate Summations
Now, we will calculate the required summations:

\[ \Sigma f_i = 12 + 14 + 65 + 107 + 157 + 202 + 222 + 230 = 1025 \]

\[ \Sigma f_i u_i = -48 - 42 - 130 - 107 + 0 + 202 + 444 + 690 = 1009 \]

### Step 7: Calculate Mean (x̄)
Using the formula for mean:

\[ \bar{x} = A + \frac{\Sigma f_i u_i}{\Sigma f_i} \cdot h \]

Substituting the values:

\[ \bar{x} = 75 + \frac{1009}{1025} \cdot 15 \]

Calculating:

\[ \bar{x} = 75 + 14.77 \approx 89.77 \]

### Step 8: Calculate Variance and Standard Deviation
Now, we need to calculate \(f_i u_i^2\):

| u_i | f_i * u_i^2 |
|------|--------------|
| -4 | 192 |
| -3 | 126 |
| -2 | 260 |
| -1 | 107 |
| 0 | 0 |
| 1 | 202 |
| 2 | 888 |
| 3 | 2070 |

Calculating \(\Sigma f_i u_i^2\):

\[ \Sigma f_i u_i^2 = 192 + 126 + 260 + 107 + 0 + 202 + 888 + 2070 = 2885 \]

Using the formula for variance:

\[ \sigma^2 = \frac{\Sigma f_i u_i^2}{\Sigma f_i} - \left(\frac{\Sigma f_i u_i}{\Sigma f_i}\right)^2 \cdot h^2 \]

Substituting the values:

\[ \sigma^2 = \frac{2885}{1025} - \left(\frac{1009}{1025}\right)^2 \cdot 15^2 \]

Calculating:

\[ \sigma^2 = 2.81 - (0.983)^2 \cdot 225 \approx 2.81 - 216.09 \approx 25.25 \]

Taking the square root to find the standard deviation:

\[ \sigma \approx \sqrt{25.25} \approx 5.02 \]

### Final Results
- Mean (x̄) ≈ 89.77
- Standard Deviation (σ) ≈ 5.02