<p>The mean and the standard deviation (s.d.) of five observations are 9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is</p>
Step-by-Step Solution
Key Concept: When standard deviation is 0, all observations are identical. Changing one observation breaks this uniformity, and the new s.d. depends on how far that changed observation deviates from the new mean.
<p><strong>Step 1:</strong> Since s.d. = 0 for the five observations with mean 9, all observations must be equal to 9.</p><p>Initial set: {9, 9, 9, 9, 9}</p><p><strong>Step 2:</strong> Let the changed observation be x. The new set has mean 10, so:<br/>Σ(new observations) = 50<br/>(9 + 9 + 9 + 9 + x) = 50<br/>36 + x = 50<br/>x = 14</p><p><strong>Step 3:</strong> New set: {9, 9, 9, 9, 14} with mean = 10</p><p><strong>Step 4:</strong> Calculate variance:<br/>Variance = Σ(xᵢ - mean)²/n<br/>= [(9-10)² + (9-10)² + (9-10)² + (9-10)² + (14-10)²]/5<br/>= [1 + 1 + 1 + 1 + 16]/5<br/>= 20/5 = 4</p><p><strong>Step 5:</strong> Standard deviation = √4 = 2</p><p>∴ Answer: C (s.d. = 2)</p>
Correct Answer: C