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

Options

A. 5.5
B. 3.87
C. 2.97
D. 2.87

Explanation

To find the standard deviation of the first 10 natural numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), we can 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 (\(\mu\))
The mean is calculated using the formula:
\[ \mu = \frac{\sum x_i}{n} \]
Where \(n\) is the number of observations (in this case, 10).

Calculating the sum:
\[ \sum x_i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 \]
Now, substitute into the mean formula:
\[ \mu = \frac{55}{10} = 5.5 \]

### Step 3: Calculate the sum of squares of the numbers (\(\sum x_i^2\))
We need to calculate:
\[ \sum x_i^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 \]
Calculating each square:
\[ 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385 \]

### Step 4: Calculate the variance (\(\sigma^2\))
The variance is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \mu^2 \]
Substituting the values:
\[ \sigma^2 = \frac{385}{10} - (5.5)^2 \]
Calculating:
\[ \sigma^2 = 38.5 - 30.25 = 8.25 \]

### Step 5: Calculate the standard deviation (\(\sigma\))
The standard deviation 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:
\[ \text{Standard Deviation} = 2.87 \]

### Conclusion
The correct answer is option (d) 2.87.

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