mean deviation about median for the following data 2,5,7,2,4,6,10,12,5

Options

A. 2.4
B. 3
C. 1.9
D. None of these

Answer

To find the mean deviation about the median for the given data set \(2, 5, 7, 2, 4, 6, 10, 12, 5\), we will follow these steps:

### Step 1: Organize the Data
First, we need to arrange the data in ascending order.

**Data in ascending order:**
\[2, 2, 4, 5, 5, 6, 7, 10, 12\]

### Step 2: Create a Frequency Distribution Table
Next, we will create a frequency distribution table.

| \(x_i\) | Frequency (\(f_i\)) |
|---------|---------------------|
| 2 | 2 |
| 4 | 1 |
| 5 | 2 |
| 6 | 1 |
| 7 | 1 |
| 10 | 1 |
| 12 | 1 |

### Step 3: Calculate Cumulative Frequency
Now, we calculate the cumulative frequency.

| \(x_i\) | Frequency (\(f_i\)) | Cumulative Frequency (\(cf\)) |
|---------|---------------------|-------------------------------|
| 2 | 2 | 2 |
| 4 | 1 | 3 |
| 5 | 2 | 5 |
| 6 | 1 | 6 |
| 7 | 1 | 7 |
| 10 | 1 | 8 |
| 12 | 1 | 9 |

### Step 4: Find the Median
To find the median, we need to find \( \frac{N}{2} \) where \( N \) is the total number of observations. Here, \( N = 9 \), so \( \frac{N}{2} = 4.5 \).

Looking at the cumulative frequency, the median is the value corresponding to the cumulative frequency just greater than \( 4.5 \), which is \( 5 \). Thus, the median is:

**Median = 5**

### Step 5: Calculate Deviations from the Median
Now, we calculate the absolute deviations from the median.

\[ \text{Deviation} = |x_i - \text{Median}| \]

| \(x_i\) | Deviation (\(|x_i - 5|\)) |
|---------|---------------------------|
| 2 | 3 |
| 2 | 3 |
| 4 | 1 |
| 5 | 0 |
| 5 | 0 |
| 6 | 1 |
| 7 | 2 |
| 10 | 5 |
| 12 | 7 |

### Step 6: Calculate \( f_i \cdot d_i \)
Now, we multiply each deviation by its corresponding frequency.

| \(x_i\) | Frequency (\(f_i\)) | Deviation (\(|x_i - 5|\)) | \(f_i \cdot d_i\) |
|---------|---------------------|---------------------------|--------------------|
| 2 | 2 | 3 | 6 |
| 4 | 1 | 1 | 1 |
| 5 | 2 | 0 | 0 |
| 6 | 1 | 1 | 1 |
| 7 | 1 | 2 | 2 |
| 10 | 1 | 5 | 5 |
| 12 | 1 | 7 | 7 |

### Step 7: Sum of \( f_i \cdot d_i \) and \( f_i \)
Now, we calculate the total of \( f_i \cdot d_i \) and \( f_i \).

\[ \sum f_i \cdot d_i = 6 + 1 + 0 + 1 + 2 + 5 + 7 = 22 \]
\[ \sum f_i = 9 \]

### Step 8: Calculate Mean Deviation
Finally, we calculate the mean deviation about the median using the formula:

\[ \text{Mean Deviation} = \frac{\sum f_i \cdot d_i}{\sum f_i} = \frac{22}{9} \approx 2.44 \]

### Final Answer
The mean deviation about the median is approximately **2.44**.

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