Consider the numbers 1,2,3,4,5,6,7,8,9,10. If 2 is added to each number then variance of the numbers so obtained is
(i) 6.5
(ii) 2.87
(iii) 3.87
(iv) 8.25

Options

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

Answer

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

### Step 1: Calculate the original numbers' sum
The original numbers are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \]

The sum of these numbers can be calculated using the formula for the sum of the first n natural numbers:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
where \( n = 10 \):
\[ \text{Sum} = \frac{10(10 + 1)}{2} = \frac{10 \times 11}{2} = 55 \]

### Step 2: Add 2 to each number
When we add 2 to each number, the new set of numbers becomes:
\[ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \]

### Step 3: Calculate the new sum
The sum of the new numbers is:
\[ \text{New Sum} = 55 + 2 \times 10 = 55 + 20 = 75 \]

### Step 4: Calculate the sum of squares of the new numbers
Now, we need to calculate the sum of the squares of the new numbers:
\[ 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 \]
Calculating each square:
\[ = 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144 \]
Adding these values:
\[ = 9 + 16 = 25 \]
\[ = 25 + 25 = 50 \]
\[ = 50 + 36 = 86 \]
\[ = 86 + 49 = 135 \]
\[ = 135 + 64 = 199 \]
\[ = 199 + 81 = 280 \]
\[ = 280 + 100 = 380 \]
\[ = 380 + 121 = 501 \]
\[ = 501 + 144 = 645 \]

### Step 5: Calculate the variance
The variance \( \sigma^2 \) is given by the formula:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Where:
- \( \sum x_i^2 = 645 \)
- \( n = 10 \)
- \( \sum x_i = 75 \)

Calculating the variance:
\[ \sigma^2 = \frac{645}{10} - \left(\frac{75}{10}\right)^2 \]
Calculating each term:
\[ = 64.5 - \left(7.5\right)^2 \]
\[ = 64.5 - 56.25 \]
\[ = 8.25 \]

### Final Answer
Thus, the variance of the numbers obtained after adding 2 to each number is:
\[ \boxed{8.25} \]