If 1 is added to first 10 natural numbers, then variance of the numbers so obtained is
Show Hint
Never waste time adding constants and recalculating means during exams! Adding a constant shifts the whole data block but keeps the internal gaps identical, so simply calculate $\frac{n^2-1}{12}$ for the original count.
Step 1: Understanding the Question:
The question asks for the statistical variance of a modified data set created by adding a constant numerical value (1) to each of the first 10 natural numbers ($1, 2, 3, \dots, 10$).
Step 2: Key Formula or Approach:
A fundamental mathematical property of variance states that variance is entirely independent of a change of origin. Adding or subtracting a constant value $k$ to every single data point in a set leaves the final spread and variance completely unchanged:
$$ \text{Var}(X + k) = \text{Var}(X) $$
Therefore, the variance of the numbers $2, 3, 4, \dots, 11$ is exactly equal to the variance of the original first 10 natural numbers. The variance of the first $n$ natural numbers is given by the standard formula:
$$ \text{Var} = \frac{n^2 - 1}{12} $$
Step 3: Detailed Explanation:
Given that we are dealing with the first $n = 10$ natural numbers, we can apply the formula directly:
$$ \text{Var} = \frac{10^2 - 1}{12} $$
$$ \text{Var} = \frac{100 - 1}{12} = \frac{99}{12} $$
Dividing both the numerator and denominator by 3:
$$ \text{Var} = \frac{33}{4} = 8.25 $$
Step 4: Final Answer:
The variance of the obtained numbers is 8.25, corresponding to option (A).
