The variance for first 20 natural numbers is

Options

A. `133//4`
B. `279//12`
C. `133//2`
D. `399//4`

Answer

To find the variance of the first 20 natural numbers, we can follow these steps:

### Step 1: Understand the Formula for Variance
The variance \( \sigma^2 \) of a set of numbers is given by the formula:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
where \( \sum x_i \) is the sum of the numbers, \( \sum x_i^2 \) is the sum of the squares of the numbers, and \( n \) is the number of elements.

### Step 2: Calculate the Sum of the First 20 Natural Numbers
The sum of the first \( n \) natural numbers is given by the formula:
\[ \sum x_i = \frac{n(n + 1)}{2} \]
For \( n = 20 \):
\[ \sum x_i = \frac{20(20 + 1)}{2} = \frac{20 \times 21}{2} = 210 \]

### Step 3: Calculate the Sum of the Squares of the First 20 Natural Numbers
The sum of the squares of the first \( n \) natural numbers is given by the formula:
\[ \sum x_i^2 = \frac{n(n + 1)(2n + 1)}{6} \]
For \( n = 20 \):
\[ \sum x_i^2 = \frac{20(20 + 1)(2 \times 20 + 1)}{6} = \frac{20 \times 21 \times 41}{6} \]
Calculating this:
\[ = \frac{20 \times 21 \times 41}{6} = \frac{17220}{6} = 2870 \]

### Step 4: Calculate the Mean
Now, we can calculate the mean \( \mu \):
\[ \mu = \frac{\sum x_i}{n} = \frac{210}{20} = 10.5 \]

### Step 5: Calculate the Variance
Now we can substitute the values into the variance formula:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Substituting the known values:
\[ \sigma^2 = \frac{2870}{20} - (10.5)^2 \]
Calculating \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]
Now substituting this back:
\[ \sigma^2 = 143.5 - 110.25 = 33.25 \]

### Step 6: Final Result
Thus, the variance of the first 20 natural numbers is:
\[ \sigma^2 = 33.25 \]