The standard deviation of first 10 natural numbers is 8.25             (b) 6.5           (c) 3.87       (d) 2.87

Options

A. (a) 8.25
B. (b) 6.5
C. (c) 3.87
D. (d) 2.87

Explanation

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

### Step 1: List 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
The mean (\( \bar{x} \)) is calculated using the formula:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]
Where \( n \) is the total number of observations.

Here, \( n = 10 \) and the sum of the first 10 natural numbers is:
\[ \sum_{i=1}^{10} x_i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 \]

Thus, the mean is:
\[ \bar{x} = \frac{55}{10} = 5.5 \]

### Step 3: Calculate the deviations from the mean
Next, we calculate the deviation of each number from the mean:
- \( 1 - 5.5 = -4.5 \)
- \( 2 - 5.5 = -3.5 \)
- \( 3 - 5.5 = -2.5 \)
- \( 4 - 5.5 = -1.5 \)
- \( 5 - 5.5 = -0.5 \)
- \( 6 - 5.5 = 0.5 \)
- \( 7 - 5.5 = 1.5 \)
- \( 8 - 5.5 = 2.5 \)
- \( 9 - 5.5 = 3.5 \)
- \( 10 - 5.5 = 4.5 \)

### Step 4: Calculate the squared deviations
Now, we square each of the deviations:
- \( (-4.5)^2 = 20.25 \)
- \( (-3.5)^2 = 12.25 \)
- \( (-2.5)^2 = 6.25 \)
- \( (-1.5)^2 = 2.25 \)
- \( (-0.5)^2 = 0.25 \)
- \( (0.5)^2 = 0.25 \)
- \( (1.5)^2 = 2.25 \)
- \( (2.5)^2 = 6.25 \)
- \( (3.5)^2 = 12.25 \)
- \( (4.5)^2 = 20.25 \)

### Step 5: Calculate the variance
The variance (\( \sigma^2 \)) is the average of these squared deviations:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} \]
Calculating the sum of squared deviations:
\[ 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 82.5 \]

Now, we divide by \( n \):
\[ \sigma^2 = \frac{82.5}{10} = 8.25 \]

### Step 6: Calculate the standard deviation
The standard deviation (\( \sigma \)) is the square root of the variance:
\[ \sigma = \sqrt{8.25} \approx 2.87 \]

### Final Answer
Thus, the standard deviation of the first 10 natural numbers is:
\[ \sigma \approx 2.87 \]