Step 1: Understanding the Concept:
The standard deviation (\( \sigma \)) of a set of data is independent of the origin. Shifting the data by subtracting a constant (like 3) does not change the standard deviation. Step 2: Key Formula or Approach:
Let \( y_i = x_i - 3 \). Then standard deviation is:
\[ \sigma = \sqrt{\frac{\sum y_i^2}{n} - \left(\frac{\sum y_i}{n}\right)^2} \] Step 3: Detailed Explanation:
Given \( n = 10 \), \( \sum y_i = 7 \), and \( \sum y_i^2 = 27 \).
\[ \sigma = \sqrt{\frac{27}{10} - \left(\frac{7}{10}\right)^2} \]
\[ \sigma = \sqrt{2.7 - (0.7)^2} = \sqrt{2.7 - 0.49} \]
\[ \sigma = \sqrt{2.21} \]
Calculating the square root:
\( \sqrt{2.21} \approx 1.4866 \). Step 4: Final Answer:
The standard deviation is 1.486.