Question

The variance of 1,2,3,4,5,6,7,8,9 and 10 is

• A. $\frac{101}{12}$
• B. $\frac{11}{12}$
• C. $\frac{99}{12}$
• D. $\frac{9}{12}$

Hint:

Using the standard formula $\sigma^2 = \frac{n^2 - 1}{12}$ is much faster than calculating the mean and summing individual squared deviations for consecutive integer series.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The given sequence represents the first $n$ natural numbers.
We can compute the variance of these consecutive integers using general algebraic series formulas.
Key Formula or Approach:
The variance of the first $n$ natural numbers is given by the formula:
\[ \sigma^2 = \frac{n^2 - 1}{12} \]

Step 2: Detailed Explanation:
The dataset contains the first 10 natural numbers: $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
Here, the number of observations is $n = 10$.
Substitute $n = 10$ into the variance formula:
\[ \sigma^2 = \frac{10^2 - 1}{12} \]
\[ \sigma^2 = \frac{100 - 1}{12} = \frac{99}{12} \]
This fraction simplifies to $8.25$.
Comparing with the given choices, the value is represented in fractional form as $\frac{99}{12}$.
Therefore, the variance is $\frac{99}{12}$.

Step 3: Final Answer
The correct option is (C).