Consider the Observations `x_1=1,x_2=2,x_3=3,.....x_100=100,x_(101)=101,x_(102)=102,x_(103)=103,x_(104)=104`
Mean deviation from the median of the given data is

Options

A. 51/2
B. 26
C. `(51 xx 52)/(103)`
D. None of these

Answer

To find the mean deviation from the median of the given observations \( x_1 = 1, x_2 = 2, \ldots, x_{104} = 104 \), we will follow these steps:

### Step 1: Calculate the Median
1. **Identify the number of observations (N)**:
\[ N = 104 \]
Since \( N \) is even, the median is calculated using the formula:
\[ \text{Median} = \frac{x_{(N/2)} + x_{(N/2 + 1)}}{2} \]
Here, \( N/2 = 52 \) and \( N/2 + 1 = 53 \).

2. **Find the values of \( x_{52} \) and \( x_{53} \)**:
\[ x_{52} = 52, \quad x_{53} = 53 \]

3. **Calculate the median**:
\[ \text{Median} = \frac{52 + 53}{2} = \frac{105}{2} = 52.5 \]

### Step 2: Calculate the Mean Deviation from the Median
1. **Use the formula for mean deviation**:
\[ \text{Mean Deviation} = \frac{\sum |x_i - m|}{N} \]
where \( m \) is the median.

2. **Calculate \( |x_i - m| \) for each observation**:
- For \( x_1 = 1 \):
\[ |1 - 52.5| = 51.5 \]
- For \( x_2 = 2 \):
\[ |2 - 52.5| = 50.5 \]
- Continue this process up to \( x_{52} \):
\[ |52 - 52.5| = 0.5 \]
- For \( x_{53} = 53 \):
\[ |53 - 52.5| = 0.5 \]
- For \( x_{54} = 54 \):
\[ |54 - 52.5| = 1.5 \]
- Continue this process up to \( x_{104} = 104 \):
\[ |104 - 52.5| = 51.5 \]

3. **Sum the absolute deviations**:
- From \( x_1 \) to \( x_{52} \):
\[ \sum_{i=1}^{52} |x_i - 52.5| = 51.5 + 50.5 + \ldots + 0.5 = \sum_{k=0}^{51} (51.5 - k) = 52 \times 51.5 - \sum_{k=0}^{51} k \]
- The sum of the first 51 natural numbers:
\[ \sum_{k=0}^{51} k = \frac{51 \times 52}{2} = 1326 \]
- Therefore:
\[ \sum_{i=1}^{52} |x_i - 52.5| = 52 \times 51.5 - 1326 = 2678 - 1326 = 1352 \]

- From \( x_{53} \) to \( x_{104} \):
\[ \sum_{i=53}^{104} |x_i - 52.5| = 0.5 + 1.5 + \ldots + 51.5 = \sum_{k=0}^{51} (k + 0.5) = \sum_{k=0}^{51} k + 52 \times 0.5 \]
- The sum is:
\[ 1326 + 26 = 1352 \]

4. **Total absolute deviation**:
\[ \text{Total} = 1352 + 1352 = 2704 \]

5. **Calculate the mean deviation**:
\[ \text{Mean Deviation} = \frac{2704}{104} = 26 \]

### Final Answer
The mean deviation from the median of the given data is \( \boxed{26} \).