Standard deviation of first n odd natural numbers is

• A
• √(n(n + 2)) / 3
• √(n² − 1) / 3
• n

The standard deviation of the first n odd natural numbers can be derived using the formula for standard deviation, which measures the amount of variation or dispersion in a set of values.

Understanding Odd Natural Numbers

The first n odd natural numbers are: 1, 3, 5, ..., (2n - 1). The sum of these numbers can be calculated using the formula:

• Sum = n²

Mean Calculation

The mean (average) of these numbers is given by:

• Mean = (Sum of numbers) / n = n² / n = n

Variance and Standard Deviation

To find the variance, we first calculate the squared differences from the mean:

• Variance = [(1 - n)² + (3 - n)² + ... + ((2n - 1) - n)²] / n

This simplifies to:

• Variance = (n² - 1) / 3

The standard deviation is the square root of the variance:

• Standard Deviation = √[(n² - 1) / 3]

Final Expression

Thus, the standard deviation of the first n odd natural numbers can be expressed as:

• Standard Deviation = A √(n(n + 2)) / 3 √(n² − 1) / 3 n

This formula provides a clear way to calculate the standard deviation based on the number of odd natural numbers you are considering.