The variance of first 5 natural numbers is
(i) 1
(ii) 2
(iii) 3
(iv) 4

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the variance of the first 5 natural numbers, we can follow these steps:

### Step 1: Identify the first 5 natural numbers
The first 5 natural numbers are:
\[ 1, 2, 3, 4, 5 \]

### Step 2: Calculate the sum of the first 5 natural numbers
We can use the formula for the sum of the first \( n \) natural numbers:
\[ S_n = \frac{n(n + 1)}{2} \]
For \( n = 5 \):
\[ S_5 = \frac{5(5 + 1)}{2} = \frac{5 \times 6}{2} = 15 \]

### Step 3: Calculate the mean of the first 5 natural numbers
The mean (\( \bar{x} \)) is calculated by dividing the sum by the number of values:
\[ \bar{x} = \frac{S}{n} = \frac{15}{5} = 3 \]

### Step 4: Calculate the variance
Variance (\( \sigma^2 \)) is defined as the average of the squared differences from the Mean:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]
Where \( x_i \) are the individual numbers.

Calculating each squared difference:
- For \( x_1 = 1 \): \( (1 - 3)^2 = (-2)^2 = 4 \)
- For \( x_2 = 2 \): \( (2 - 3)^2 = (-1)^2 = 1 \)
- For \( x_3 = 3 \): \( (3 - 3)^2 = (0)^2 = 0 \)
- For \( x_4 = 4 \): \( (4 - 3)^2 = (1)^2 = 1 \)
- For \( x_5 = 5 \): \( (5 - 3)^2 = (2)^2 = 4 \)

Now, sum these squared differences:
\[ 4 + 1 + 0 + 1 + 4 = 10 \]

Now, divide by the number of values (5):
\[ \sigma^2 = \frac{10}{5} = 2 \]

### Final Answer
The variance of the first 5 natural numbers is \( 2 \).