Find the standard deviation of the first five even natural numbers.

Answer

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

### Step 1: Identify the first five even natural numbers
The first five even natural numbers are:
- 2, 4, 6, 8, 10

### Step 2: Calculate the mean (average) of these numbers
To find the mean, we sum the numbers and divide by the count of numbers.

\[ \text{Mean} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6 \]

### Step 3: Calculate the deviations from the mean
Next, we find the deviation of each number from the mean:

- Deviation for 2: \(2 - 6 = -4\)
- Deviation for 4: \(4 - 6 = -2\)
- Deviation for 6: \(6 - 6 = 0\)
- Deviation for 8: \(8 - 6 = 2\)
- Deviation for 10: \(10 - 6 = 4\)

### Step 4: Square each deviation
Now, we square each of the deviations:

- \((-4)^2 = 16\)
- \((-2)^2 = 4\)
- \(0^2 = 0\)
- \(2^2 = 4\)
- \(4^2 = 16\)

### Step 5: Calculate the variance
To find the variance, we take the average of these squared deviations:

\[ \text{Variance} = \frac{16 + 4 + 0 + 4 + 16}{5} = \frac{40}{5} = 8 \]

### Step 6: Calculate the standard deviation
The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sqrt{8} = 2\sqrt{2} \]

### Conclusion
Thus, the standard deviation of the first five even natural numbers is:

\[ \boxed{2\sqrt{2}} \]

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