The standard deviation for first 10 natural number is

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, we can 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:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]
where \( n \) is the number of observations (in this case, 10).

Calculating the sum:
\[ \sum_{i=1}^{10} x_i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 \]
Now, substituting into the mean formula:
\[ \bar{x} = \frac{55}{10} = 5.5 \]

### Step 3: Calculate the variance (\( \sigma^2 \))
The variance is calculated using the formula:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
First, we need to calculate \( (x_i - \bar{x})^2 \) for each \( x_i \):
- For \( 1 \): \( (1 - 5.5)^2 = 20.25 \)
- For \( 2 \): \( (2 - 5.5)^2 = 12.25 \)
- For \( 3 \): \( (3 - 5.5)^2 = 6.25 \)
- For \( 4 \): \( (4 - 5.5)^2 = 2.25 \)
- For \( 5 \): \( (5 - 5.5)^2 = 0.25 \)
- For \( 6 \): \( (6 - 5.5)^2 = 0.25 \)
- For \( 7 \): \( (7 - 5.5)^2 = 2.25 \)
- For \( 8 \): \( (8 - 5.5)^2 = 6.25 \)
- For \( 9 \): \( (9 - 5.5)^2 = 12.25 \)
- For \( 10 \): \( (10 - 5.5)^2 = 20.25 \)

Now, summing these squared differences:
\[ \sum_{i=1}^{10} (x_i - \bar{x})^2 = 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 72.5 \]

Now, substituting into the variance formula:
\[ \sigma^2 = \frac{72.5}{10} = 7.25 \]

### Step 4: Calculate the standard deviation (\( \sigma \))
The standard deviation is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{7.25} \approx 2.69 \]

### Final Answer
The standard deviation of the first 10 natural numbers is approximately \( 2.69 \).

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