For a frequency distribution the mean deviation about mean is computed by

Options

A. M.D. `=(sum d_(1))/(sum f_(i))`
B. `M.D. =(sum f_(i)d_(i))/(sum f_(i))`
C. `M.D.=(sum f_(i) |d_(i)|)/(sum f_(i))`
D. `M.D.=(sum f_(i))/(sum f_(i) |d_(i)|)`

Answer

To compute the mean deviation about the mean for a frequency distribution, we can follow these steps:

### Step 1: Understand the Components
We need to identify the components involved in the calculation:
- \( x_i \): The values in the dataset.
- \( f_i \): The frequency of each value \( x_i \).
- \( \bar{x} \): The mean of the dataset.

### Step 2: Calculate the Mean
The mean \( \bar{x} \) is calculated using the formula:
\[ \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \]
This gives us the average value of the dataset, weighted by the frequencies.

### Step 3: Calculate the Deviations
Next, we need to calculate the deviations from the mean:
\[ d_i = x_i - \bar{x} \]
Here, \( d_i \) represents the deviation of each value from the mean.

### Step 4: Take Absolute Values
Since we are interested in the mean deviation, we take the absolute values of the deviations:
\[ |d_i| = |x_i - \bar{x}| \]

### Step 5: Multiply by Frequencies
Now, we multiply each absolute deviation by its corresponding frequency:
\[ f_i \cdot |d_i| = f_i \cdot |x_i - \bar{x}| \]

### Step 6: Sum the Products
We then sum these products over all values:
\[ \sum (f_i \cdot |d_i|) = \sum (f_i \cdot |x_i - \bar{x}|) \]

### Step 7: Divide by Total Frequency
Finally, we divide the total sum of the products by the total frequency to get the mean deviation:
\[ \text{Mean Deviation} = \frac{\sum (f_i \cdot |d_i|)}{\sum f_i} \]

### Final Formula
Thus, the formula for the mean deviation about the mean for a frequency distribution is:
\[ \text{Mean Deviation} = \frac{\sum (f_i \cdot |x_i - \bar{x}|)}{\sum f_i} \]