<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 observations shifts measures of central tendency and position equally, but measures of spread (dispersion) remain unchanged because they depend on deviations from the mean, which don't change when all values shift by the same amount.
<p><strong>Step 1:</strong> Understand the effect of adding a constant c to all observations:</p><p>If we add constant c to every observation:</p><ul><li>Mean increases by c: μ' = μ + c</li><li>Median increases by c: M' = M + c</li><li>Mode increases by c: Mo' = Mo + c</li></ul><p><strong>Step 2:</strong> Analyze measures of dispersion:</p><p>For variance: Each (xᵢ - mean) becomes (xᵢ + 10) - (mean + 10) = (xᵢ - mean)</p><p>The deviations from mean remain identical, so:</p><ul><li>Variance σ² remains unchanged</li><li>Standard deviation σ remains unchanged</li><li>Range remains unchanged (both max and min increase by 10)</li><li>IQR, MAD all remain unchanged</li></ul><p><strong>Step 3:</strong> The answer depends on the options given. Measures of dispersion like <strong>standard deviation, variance, range, or quartile deviation</strong> will NOT change.</p><p>∴ Answer: <strong>C</strong> (likely standard deviation, variance, or range - whichever is the option representing a measure of dispersion)</p>
Correct Answer: C