Find the Variance of First 10 Natural Numbers.

Answer

To find the variance of the first 10 natural numbers, we will follow these steps:

### Step 1: Identify the first 10 natural numbers
The first 10 natural numbers are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \]

### Step 2: Calculate the mean (\( \bar{x} \))
The mean is calculated using the formula:
\[ \bar{x} = \frac{\text{Sum of all values}}{n} \]
Where \( n \) is the number of values. The sum of the first \( n \) natural numbers can be calculated using the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
For \( n = 10 \):
\[ \text{Sum} = \frac{10 \times (10 + 1)}{2} = \frac{10 \times 11}{2} = 55 \]
Now, we can find the mean:
\[ \bar{x} = \frac{55}{10} = 5.5 \]

### Step 3: Calculate the variance
Variance is calculated using the formula:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
Where \( x_i \) represents each value in the dataset. We will calculate \( (x_i - \bar{x})^2 \) for each \( x_i \).

Calculating \( (x_i - \bar{x})^2 \) for each \( x_i \):
- For \( x_1 = 1 \): \( (1 - 5.5)^2 = (-4.5)^2 = 20.25 \)
- For \( x_2 = 2 \): \( (2 - 5.5)^2 = (-3.5)^2 = 12.25 \)
- For \( x_3 = 3 \): \( (3 - 5.5)^2 = (-2.5)^2 = 6.25 \)
- For \( x_4 = 4 \): \( (4 - 5.5)^2 = (-1.5)^2 = 2.25 \)
- For \( x_5 = 5 \): \( (5 - 5.5)^2 = (-0.5)^2 = 0.25 \)
- For \( x_6 = 6 \): \( (6 - 5.5)^2 = (0.5)^2 = 0.25 \)
- For \( x_7 = 7 \): \( (7 - 5.5)^2 = (1.5)^2 = 2.25 \)
- For \( x_8 = 8 \): \( (8 - 5.5)^2 = (2.5)^2 = 6.25 \)
- For \( x_9 = 9 \): \( (9 - 5.5)^2 = (3.5)^2 = 12.25 \)
- For \( x_{10} = 10 \): \( (10 - 5.5)^2 = (4.5)^2 = 20.25 \)

### Step 4: Sum the squared differences
Now we sum all the squared differences:
\[ \sum (x_i - \bar{x})^2 = 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 92.5 \]

### Step 5: Calculate the variance
Now we can calculate the variance:
\[ \sigma^2 = \frac{1}{10} \times 92.5 = 9.25 \]

### Final Answer
The variance of the first 10 natural numbers is:
\[ \boxed{9.25} \]