Statistics

To solve this problem, we are given 50 observations with a variance of 7, and we are tasked with finding the variance of a transformed dataset.

1. Understanding the Variance Formula:
The variance is given by the formula:

where is the mean of the observations. In this case, we are told that , so we can write:

2. Transformation of Data:
We now transform each observation by multiplying by 6 and subtracting 5. The new observations are given by:

Let be the mean of the new observations. Then, we can find as:

3. Variance of the New Data:
The variance of the new data, , is given by:

Substitute the expression for and :

Since , we get:

Final Answer:
The variance of the new data is .

We are given the following set of numbers:

Step 1: Compute the Mean Deviation from the Mean ( ) The mean deviation from the mean is given by: where represents the mean of the dataset, and are the individual values. First, we calculate : Now, compute :

Step 2: Compute the Mean Deviation from the Median ( ) To determine , we first find the median by arranging the data in ascending order: Since there are 9 values, the median is the middle value: Now, compute :

Step 3: Compute The difference between the mean deviation from the mean and the mean deviation from the median is: Approximating this value: Thus, the final result is:

First, calculate the mean of the data: Next, compute the sum of the squared differences from the mean: Finally, the variance ( ) is calculated as:

Step 1: Identify the data set
The first 10 natural numbers that are multiples of 3 are: This forms an arithmetic sequence where: - First term ( ) = 3 - Common difference ( ) = 3 - Number of terms ( ) = 10

Step 2: Compute the mean ( )
The mean of a sequence is given by: Using the sum formula for an arithmetic sequence:

Step 3: Compute the variance ( )
Variance is given by: Computing the squared differences: Summing these: Thus, the correct answer is:

Step 1: Class midpoints and frequencies.
For each class interval, the midpoint is: Frequencies are 2, 3, 2, 1 respectively. The total frequency

Step 2: Calculate the mean .

Step 3: Compute and then variance.
Hence The variance is Therefore, the variance is