Find the mean deviation from mean for the following data :
`{:(x_i:, 5,10,15,20,25),(f_i:,7,4,6,3,5):}`

Answer

To find the mean deviation from the mean for the given data, we will follow these steps:

### Step 1: Calculate the total frequency (Σf)
The frequencies given are:
- \( f_1 = 7 \)
- \( f_2 = 4 \)
- \( f_3 = 6 \)
- \( f_4 = 3 \)
- \( f_5 = 5 \)

Now, we will calculate the total frequency:
\[ \Sigma f = 7 + 4 + 6 + 3 + 5 = 25 \]

### Step 2: Calculate Σ(x_i * f_i)
Next, we will calculate the product of each \( x_i \) and its corresponding \( f_i \):
- \( x_1 = 5 \), \( f_1 = 7 \) → \( x_1 * f_1 = 5 * 7 = 35 \)
- \( x_2 = 10 \), \( f_2 = 4 \) → \( x_2 * f_2 = 10 * 4 = 40 \)
- \( x_3 = 15 \), \( f_3 = 6 \) → \( x_3 * f_3 = 15 * 6 = 90 \)
- \( x_4 = 20 \), \( f_4 = 3 \) → \( x_4 * f_4 = 20 * 3 = 60 \)
- \( x_5 = 25 \), \( f_5 = 5 \) → \( x_5 * f_5 = 25 * 5 = 125 \)

Now, we will sum these products:
\[ \Sigma (x_i * f_i) = 35 + 40 + 90 + 60 + 125 = 350 \]

### Step 3: Calculate the mean (x̄)
Using the formula for the mean:
\[ \bar{x} = \frac{\Sigma (x_i * f_i)}{\Sigma f} = \frac{350}{25} = 14 \]

### Step 4: Calculate the absolute deviations |x_i - x̄|
Now, we will calculate the absolute deviations from the mean:
- For \( x_1 = 5 \): \( |5 - 14| = 9 \)
- For \( x_2 = 10 \): \( |10 - 14| = 4 \)
- For \( x_3 = 15 \): \( |15 - 14| = 1 \)
- For \( x_4 = 20 \): \( |20 - 14| = 6 \)
- For \( x_5 = 25 \): \( |25 - 14| = 11 \)

### Step 5: Multiply the absolute deviations by their corresponding frequencies
Now, we will multiply each absolute deviation by its corresponding frequency:
- For \( x_1 = 5 \): \( 7 * 9 = 63 \)
- For \( x_2 = 10 \): \( 4 * 4 = 16 \)
- For \( x_3 = 15 \): \( 6 * 1 = 6 \)
- For \( x_4 = 20 \): \( 3 * 6 = 18 \)
- For \( x_5 = 25 \): \( 5 * 11 = 55 \)

### Step 6: Calculate the sum of these products
Now, we will sum these products:
\[ \Sigma (f_i * |x_i - \bar{x}|) = 63 + 16 + 6 + 18 + 55 = 158 \]

### Step 7: Calculate the mean deviation from the mean
Finally, we will calculate the mean deviation using the formula:
\[ \text{Mean Deviation} = \frac{\Sigma (f_i * |x_i - \bar{x}|)}{\Sigma f} = \frac{158}{25} = 6.32 \]

### Final Answer
The mean deviation from the mean is \( 6.32 \).

---