Mean Deviation

We know that

Given the variance: If each observation is multiplied by a constant , the variance changes as: If each observation is multiplied by 2, the new observations are , such that: The mean of the new observations is: Substituting and into equation (i), we get: Thus, the variance of the new observations is:

Let the items be . Now, given the condition: Using the sum of the first natural numbers:

Step 1: Understanding the Concept:
The mean deviation about the mean is the average of the absolute differences between each data point and the mean of the data set. For a data set , it is given by:

Step 2: Identifying the Sequence and Calculating the Mean:
The given data set is an arithmetic progression (AP) of odd numbers: .
• First term ( ) = 1
• Common difference ( ) = 2
• Last term ( ) = 101 To find the number of terms ( ): Since the distribution is symmetric, the mean ( ) is the middle term (the term):

Step 3: Calculating the Sum of Absolute Deviations:
We need to calculate . The deviations are: This results in the sequence: . Sum = Using the AP sum formula for 25 terms ( ):

Step 4: Final Answer
The mean deviation is: Rounding to one decimal place as per the options, we get 25.5.