Cosider the number 1,2,3,4,5,6,7,8,9 and 10.If 1 is added to each number the variance of the number so obtained is

Options

A. 6.5
B. 2.87
C. 3.87
D. 8.25

Explanation

To find the variance of the numbers obtained by adding 1 to each number in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, we can follow these steps:

### Step 1: Create the new set of numbers
When we add 1 to each number in the original set, we get:
- Original set: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- New set: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

### Step 2: Calculate the mean of the new set
The mean (average) of the new set can be calculated as follows:
\[ \text{Mean} = \frac{\text{Sum of all elements}}{\text{Number of elements}} = \frac{2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11}{10} \]
Calculating the sum:
\[ 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 65 \]
Thus, the mean is:
\[ \text{Mean} = \frac{65}{10} = 6.5 \]

### Step 3: Calculate the sum of squares of the new set
Next, we need to calculate the sum of the squares of the new set:
\[ \text{Sum of squares} = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 \]
Calculating each square:
\[ = 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 = 505 \]

### Step 4: Use the variance formula
The formula for variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Where:
- \( \sum x_i^2 = 505 \)
- \( n = 10 \)
- \( \sum x_i = 65 \)

Substituting these values into the variance formula:
\[ \sigma^2 = \frac{505}{10} - \left(\frac{65}{10}\right)^2 \]
Calculating each term:
\[ \sigma^2 = 50.5 - (6.5)^2 \]
Calculating \( (6.5)^2 \):
\[ (6.5)^2 = 42.25 \]
Thus, we have:
\[ \sigma^2 = 50.5 - 42.25 = 8.25 \]

### Final Answer
The variance of the new set of numbers obtained by adding 1 to each number is:
\[ \text{Variance} = 8.25 \]