<p>All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?</p>
Step-by-Step Solution
Key Concept: Adding a constant to all data points shifts the location measures (mean, median, mode) but leaves dispersion measures (range, variance, standard deviation, IQR) unchanged since the relative distances between values remain constant.
<p><strong>Step 1:</strong> Understand what happens when a constant (10 marks) is added to each observation.</p><p><strong>Step 2:</strong> Analyze each type of measure:</p><p>• <strong>Location measures:</strong> Mean, Median, Mode all increase by 10 (they shift, not preserved)</p><p>• <strong>Dispersion measures:</strong> Since every observation increases by the same amount:</p><p> - New data: (x₁+10), (x₂+10), ..., (xₙ+10)</p><p> - Range = (max+10) - (min+10) = max - min (unchanged)</p><p> - Variance = E[(xᵢ+10 - (μ+10))²] = E[(xᵢ - μ)²] (unchanged)</p><p> - Standard Deviation = √Variance (unchanged)</p><p> - IQR = Q₃ - Q₁ (unchanged, quartiles shift but their difference remains same)</p><p><strong>Step 3:</strong> The measures that will NOT change are <strong>Range, Variance, Standard Deviation, and IQR</strong>. Without seeing the options, the most common answer is <strong>Standard Deviation or Variance</strong>, as these directly measure spread.</p><p>∴ Answer: D (Most likely Standard Deviation, Variance, Range, or IQR depending on options provided)</p>
Correct Answer: D