Variance of 1st 10 natural nos. is m & mean of first 10 odd natural number in n then m/n is

Options

A. `(33)/(41)`
B. `(40)/(33)`
C. `(41)/(33)`
D. `(33)/(40)`

Explanation

To solve the problem, we need to find the variance of the first 10 natural numbers and the mean of the first 10 odd natural numbers, and then compute the ratio \( \frac{m}{n} \).

### Step 1: Calculate the Variance of the First 10 Natural Numbers

1. **Identify the first 10 natural numbers**:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \]

2. **Calculate the mean (\( \mu \)) of these numbers**:
\[ \mu = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10}{10} = \frac{55}{10} = 5.5 \]

3. **Calculate the sum of squares of these numbers**:
\[ \text{Sum of squares} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385 \]

4. **Use the variance formula**:
\[ \text{Variance} (m) = \frac{\sum x_i^2}{n} - \mu^2 \]
where \( n = 10 \).

Plugging in the values:
\[ m = \frac{385}{10} - (5.5)^2 = 38.5 - 30.25 = 8.25 \]

### Step 2: Calculate the Mean of the First 10 Odd Natural Numbers

1. **Identify the first 10 odd natural numbers**:
\[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \]

2. **Calculate the mean (\( n \)) of these numbers**:
\[ n = \frac{1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19}{10} = \frac{100}{10} = 10 \]

### Step 3: Calculate \( \frac{m}{n} \)

1. **Now, substitute the values of \( m \) and \( n \)**:
\[ \frac{m}{n} = \frac{8.25}{10} = 0.825 \]

### Final Answer:
\[ \frac{m}{n} = 0.825 \]

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