Step 1: Understanding the Concept:
The mean deviation about the mean is a measure of dispersion. It is calculated by taking the average of the absolute differences between each data point and the mean of the data set.
Step 2: Key Formula or Approach:
The formula for Mean Deviation (MD) about the mean $\bar{x}$ is:
\[ \text{MD} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}| \]
First, we must find the total number of terms $n$ and the mean $\bar{x}$ for the given arithmetic progression (AP).
Step 3: Detailed Explanation:
The given sequence is an AP: $1, 3, 5, \dots, 101$.
First term $a = 1$, common difference $d = 2$.
To find the number of terms $n$:
\[ n\text{-th term} = a + (n-1)d \implies 101 = 1 + (n-1)2 \]
\[ 100 = (n-1)2 \implies n-1 = 50 \implies n = 51 \]
Since it is an AP with an odd number of terms, the mean $\bar{x}$ is simply the middle term, or the average of the first and last terms:
\[ \bar{x} = \frac{1 + 101}{2} = 51 \]
Now, we calculate the absolute deviations $|x_i - \bar{x}|$ for each term:
For $x_i = 1, 3, 5, \dots, 49, 51, 53, \dots, 99, 101$, the mean is $51$.
Deviations are:
$|1 - 51| = 50$
$|3 - 51| = 48$
$\dots$
$|49 - 51| = 2$
$|51 - 51| = 0$
$|53 - 51| = 2$
$\dots$
$|101 - 51| = 50$
The deviations form two identical sequences from $2$ to $50$ (even numbers).
There are 25 such even numbers on either side of the mean.
Sum of absolute deviations = $2 \times (2 + 4 + 6 + \dots + 50)$
\[ = 2 \times 2(1 + 2 + 3 + \dots + 25) = 4 \times \frac{25 \times (25+1)}{2} \]
\[ = 4 \times \frac{25 \times 26}{2} = 2 \times 650 = 1300 \]
Now, divide by the total number of terms $n = 51$ to find the mean deviation:
\[ \text{MD} = \frac{1300}{51} \approx 25.4901 \approx 25.5 \]
Step 4: Final Answer:
The mean deviation about the mean is $25.5$.