Question

Standard deviation of the numbers 2,2,2,2,2,2 is

• A. 0
• B. 2
• C. $\sqrt{12}$
• D. $\frac{1}{6} \sqrt{12}$

Hint:

The standard deviation of a constant series $\{c, c, \dots, c\}$ is always zero because all values are identical to the mean.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The standard deviation measures the extent of variation or dispersion of a set of values around their arithmetic mean.
Key Formula or Approach:
The standard deviation $\sigma$ is calculated as:
\[ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \]

Step 2: Detailed Explanation:
We are given the dataset: $\{2, 2, 2, 2, 2, 2\}$.
The number of observations is $n = 6$.
Let us calculate the arithmetic mean $\bar{x}$:
\[ \bar{x} = \frac{2 + 2 + 2 + 2 + 2 + 2}{6} = \frac{12}{6} = 2 \]
Now, calculate the deviation of each observation from the mean $(x_i - \bar{x})$:
\[ x_1 - \bar{x} = 2 - 2 = 0 \]
\[ x_2 - \bar{x} = 2 - 2 = 0 \]
All deviations are equal to 0.
Summing the squared deviations yields:
\[ \sum (x_i - \bar{x})^2 = 0 + 0 + 0 + 0 + 0 + 0 = 0 \]
Substitute this sum into the standard deviation formula:
\[ \sigma = \sqrt{\frac{0}{6}} = 0 \]
Since there is no variation in the dataset, the standard deviation is 0.

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