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`
Quarter deviation of the given data is

Options

A. 26.75
B. 25
C. 26
D. None of these

Answer

To calculate the quarter deviation of the given data, we will follow these steps:

### Step 1: Identify the Observations
The observations are given as:
\[ x_1 = 1, x_2 = 2, x_3 = 3, \ldots, x_{104} = 104 \]

### Step 2: Calculate the Mean (\( \bar{x} \))
The mean is calculated using the formula:
\[ \bar{x} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

The sum of the first \( n \) natural numbers is given by:
\[ S_n = \frac{n(n + 1)}{2} \]
For \( n = 104 \):
\[ S_{104} = \frac{104 \times 105}{2} = 5460 \]

Now, the mean is:
\[ \bar{x} = \frac{5460}{104} = 52.5 \]

### Step 3: Calculate the Mean Deviation
The mean deviation is calculated using the formula:
\[ \text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n} \]

We will calculate \( |x_i - \bar{x}| \) for each observation:
- For \( x_1 = 1 \): \( |1 - 52.5| = 51.5 \)
- For \( x_2 = 2 \): \( |2 - 52.5| = 50.5 \)
- ...
- For \( x_{52} = 52 \): \( |52 - 52.5| = 0.5 \)
- For \( x_{53} = 53 \): \( |53 - 52.5| = 0.5 \)
- ...
- For \( x_{104} = 104 \): \( |104 - 52.5| = 51.5 \)

The sum of absolute deviations can be calculated as:
\[ \sum |x_i - \bar{x}| = 51.5 + 50.5 + 49.5 + \ldots + 0.5 + 0.5 + 1.5 + \ldots + 51.5 \]

This can be simplified by recognizing that the terms from \( x_1 \) to \( x_{52} \) are symmetric with respect to the mean.

Calculating the total:
- The first half contributes: \( 51.5 + 50.5 + \ldots + 0.5 \)
- The second half contributes: \( 0.5 + 1.5 + \ldots + 51.5 \)

The sum of the first 52 terms is:
\[ \sum_{k=0}^{51} k = \frac{51 \times 52}{2} = 1326 \]
The total deviation is:
\[ 2 \times 1326 = 2652 \]

Now, the mean deviation is:
\[ \text{Mean Deviation} = \frac{2652}{104} = 25.5 \]

### Step 4: Calculate the Quarter Deviation
The quarter deviation is related to the mean deviation by the formula:
\[ \text{Quarter Deviation} = \frac{5}{4} \times \text{Mean Deviation} \]
Substituting the mean deviation:
\[ \text{Quarter Deviation} = \frac{5}{4} \times 25.5 = 31.875 \]

### Final Answer
The quarter deviation of the given data is:
\[ \text{Quarter Deviation} = 31.875 \]