Consider 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 will follow these steps:

### Step 1: Identify the original set of numbers
The original set of numbers is:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \]

### Step 2: Add 1 to each number
When we add 1 to each number, we get:
\[ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \]

### Step 3: Calculate the sum of the new numbers
We need to calculate the sum of the new numbers:
\[ 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 65 \]

### Step 4: Calculate the sum of the squares of the new numbers
Next, we calculate the sum of the squares of the new numbers:
\[ 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 \]
Calculating each square:
\[ = 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 \]
Now, summing these values:
\[ = 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 = 505 \]

### Step 5: Calculate the variance
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:
- \( \sum x_i^2 = 505 \)
- \( \sum x_i = 65 \)
- \( n = 10 \) (the number of data points)

Substituting the values into the formula:
\[ \sigma^2 = \frac{505}{10} - \left(\frac{65}{10}\right)^2 \]
Calculating each term:
\[ \sigma^2 = 50.5 - (6.5)^2 \]
Calculating \( (6.5)^2 \):
\[ (6.5)^2 = 42.25 \]
Now substituting back:
\[ \sigma^2 = 50.5 - 42.25 = 8.25 \]

### Final Answer
The variance of the numbers obtained after adding 1 to each number is:
\[ \boxed{8.25} \]