Find the Standard Deviation of First 10 Even Natural numbers.

Answer

To find the standard deviation of the first 10 even natural numbers, we will follow these steps:

### Step 1: Identify the first 10 even natural numbers
The first 10 even natural numbers are:
\[ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \]

### Step 2: Calculate the mean (average)
The mean \( \bar{x} \) is calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \( n \) is the number of observations and \( x_i \) are the observations.

Calculating the sum:
\[ \sum x_i = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110 \]
The number of observations \( n = 10 \).

Now, calculate the mean:
\[ \bar{x} = \frac{110}{10} = 11 \]

### Step 3: Calculate the squared differences from the mean
Next, we calculate the squared differences from the mean for each observation:
\[ (x_i - \bar{x})^2 \]
Calculating each squared difference:
- For \( 2 \): \( (2 - 11)^2 = (-9)^2 = 81 \)
- For \( 4 \): \( (4 - 11)^2 = (-7)^2 = 49 \)
- For \( 6 \): \( (6 - 11)^2 = (-5)^2 = 25 \)
- For \( 8 \): \( (8 - 11)^2 = (-3)^2 = 9 \)
- For \( 10 \): \( (10 - 11)^2 = (-1)^2 = 1 \)
- For \( 12 \): \( (12 - 11)^2 = (1)^2 = 1 \)
- For \( 14 \): \( (14 - 11)^2 = (3)^2 = 9 \)
- For \( 16 \): \( (16 - 11)^2 = (5)^2 = 25 \)
- For \( 18 \): \( (18 - 11)^2 = (7)^2 = 49 \)
- For \( 20 \): \( (20 - 11)^2 = (9)^2 = 81 \)

### Step 4: Sum the squared differences
Now, we sum all the squared differences:
\[ \sum (x_i - \bar{x})^2 = 81 + 49 + 25 + 9 + 1 + 1 + 9 + 25 + 49 + 81 \]
Calculating this gives:
\[ \sum (x_i - \bar{x})^2 = 81 + 49 + 25 + 9 + 1 + 1 + 9 + 25 + 49 + 81 = 330 \]

### Step 5: Calculate the variance
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]
Substituting the values:
\[ \sigma^2 = \frac{330}{10} = 33 \]

### Step 6: Calculate the standard deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{33} \]

Thus, the standard deviation of the first 10 even natural numbers is:
\[ \sqrt{33} \]