Statistics
Measures of Central Tendency and Dispersion
Grade 11
Question:
<p>Statement (a): Mode can be calculated from histogram and variance is independent of origin and scale chosen, whereas median depends on origin and scale chosen.<br>Statement (b): Both (a) and (b) are true.<br><br>Which of the following is correct regarding the above statements?</p>
<p>Only (a) is true</p>
<p>Only (b) is true</p>
<p>Both (a) and (b) are true</p>
<p>Neither (a) nor (b) is true</p>
Step-by-Step Solution
Key Concept: Variance is independent of origin (additive shifts don't affect it) but dependent on scale (multiplicative changes affect it); median is dependent on both origin and scale since it's an absolute positional measure, not relative.
<p><strong>Step 1: Analyze variance properties</strong></p><p>If Y = aX + b (where a is scale, b is origin):</p><p>Var(Y) = a²Var(X) + 0</p><p>Variance is independent of origin (b term vanishes) but dependent on scale (a² factor).</p><p><strong>Step 2: Analyze median properties</strong></p><p>Median is the middle value in ordered data. Both origin shifts and scale changes affect the median value directly.</p><p>If Y = aX + b, then Median(Y) = a·Median(X) + b</p><p>Median depends on BOTH origin and scale.</p><p><strong>Step 3: Evaluate statement (a)</strong></p><p>Statement (a) claims: 'variance is independent of origin and scale' - FALSE (it's dependent on scale)</p><p>'median depends on origin and scale' - TRUE</p><p>Since one part is false, statement (a) is INCORRECT.</p><p><strong>Step 4: Evaluate statement (b)</strong></p><p>Statement (b) says 'Both (a) and (b) are true' - but statement (a) is false.</p><p>∴ Answer: C (Both statements are false or neither is completely correct)</p>
Correct Answer: C