Statistics
Mean and Standard Deviation
Grade 11
Question:
<p>The mean and the SD of 1, 2, 3, 4, 5, 6 are</p>
<p>\(\dfrac{7}{2}, \sqrt{\dfrac{35}{12}}\)</p>
<p>\(3, 3\)</p>
<p>\(\dfrac{7}{2}, \sqrt{3}\)</p>
<p>\(3, \dfrac{35}{12}\)</p>
Step-by-Step Solution
Key Concept: Mean is the arithmetic average of all values, and standard deviation measures spread by finding the average of squared deviations from the mean. For symmetric datasets around an integer mean, calculations simplify significantly.
<p><strong>Step 1: Calculate Mean</strong></p><p>Mean = (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 3.5</p><p><strong>Step 2: Calculate Deviations and their Squares</strong></p><p>Deviations from 3.5: -2.5, -1.5, -0.5, 0.5, 1.5, 2.5</p><p>Squared deviations: 6.25, 2.25, 0.25, 0.25, 2.25, 6.25</p><p><strong>Step 3: Calculate Variance</strong></p><p>Variance = (6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25)/6 = 17.5/6 = 35/12</p><p><strong>Step 4: Calculate SD</strong></p><p>SD = √(35/12) = √35/(2√3) = √105/6 ≈ 1.708</p><p><strong>Or simplifying:</strong> SD = √(35/12) = √(35/12)</p><p>∴ <strong>Mean = 3.5, SD = √(35/12) or √105/6</strong></p>
Correct Answer: A