The variance of first 50 even natural numbers is

• 1) 8 3 3 / 4
• 2) 8 3 3
• 3) 4 3 7
• 4) 4 3 7 / 4

To find the variance of the first 50 even natural numbers, we first need to identify these numbers. The first 50 even natural numbers are 2, 4, 6, ..., 100.

Calculating the Mean

The mean (average) of these numbers can be calculated as follows:

• Sum of the first 50 even numbers: 2 + 4 + 6 + ... + 100 = 2 × (1 + 2 + 3 + ... + 50) = 2 × (50 × 51 / 2) = 2550
• Mean = Total Sum / Number of Values = 2550 / 50 = 51

Finding the Variance

Variance is calculated using the formula:

Variance = (Sum of squares of each number - (Mean)^2) / Number of values

First, we find the sum of the squares of the first 50 even numbers:

• Sum of squares = 2² + 4² + 6² + ... + 100² = 4 × (1² + 2² + 3² + ... + 50²)
• Using the formula for the sum of squares: (n(n + 1)(2n + 1)) / 6, where n = 50:
• Sum of squares = 4 × (50 × 51 × 101) / 6 = 4 × 42925 / 6 = 28616.67

Now, substituting into the variance formula:

• Variance = (28616.67 - 51²) / 50
• Variance = (28616.67 - 2601) / 50 = 260.23

Final Result

The variance of the first 50 even natural numbers is approximately 260.23.

None of the provided options (1) 8 3 3 / 4, (2) 8 3 3, (3) 4 3 7, or (4) 4 3 7 / 4 match this result. Please double-check the options or the question for accuracy.