Statistics
Mean Deviation
Grade 11
Question:
<p>If each observation \(x_i\) is increased by 5 to get new observations \(x_i + 5\), then the mean deviation (MD) of the new data compared to MD of the old data is:</p>
<p>MD (New) = MD (Old) + 5</p>
<p>MD (New) = MD (Old) - 5</p>
<p>MD (New) = MD (Old)</p>
<p>MD (New) = 5 × MD (Old)</p>
Step-by-Step Solution
Key Concept: Mean deviation measures spread from the mean, which depends only on deviations between data points. Adding a constant to all observations shifts the mean by the same amount, but the deviations from the mean remain unchanged.
<p><strong>Step 1:</strong> Let original observations be x₁, x₂, ..., xₙ with mean = x̄</p><p>Original MD = (Σ|xᵢ - x̄|)/n</p><p><strong>Step 2:</strong> New observations: (x₁+5), (x₂+5), ..., (xₙ+5)</p><p>New mean = (Σ(xᵢ+5))/n = x̄ + 5</p><p><strong>Step 3:</strong> New MD = (Σ|(xᵢ+5) - (x̄+5)|)/n = (Σ|xᵢ - x̄|)/n</p><p><strong>Step 4:</strong> The absolute deviations remain identical because:</p><p>|(xᵢ+5) - (x̄+5)| = |xᵢ + 5 - x̄ - 5| = |xᵢ - x̄|</p><p><strong>∴ Answer: The new MD equals the old MD (remains unchanged)</strong></p>
Correct Answer: C