Statistics
Standard Deviation
Grade None
Question:
<p>The SD of a variate \(x\) is \(\sigma\). The SD of the variate \(\dfrac{ax+b}{c}\), where \(a, b, c\) are constants, is</p>
<p>\(\left(\dfrac{a}{c}\right)\sigma\)</p>
<p>\(\left|\dfrac{a}{c}\right|\sigma\)</p>
<p>\(\left(\dfrac{a^2}{c^2}\right)\sigma\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Standard deviation is scale-invariant to linear transformations: only the multiplicative factor (a/c) affects SD, while the additive constant (b) and location shifts have no effect on spread.
<p><strong>Step 1:</strong> Recall that for a linear transformation y = kx + m, we have SD(y) = |k|·SD(x). The additive constant m does NOT affect standard deviation since SD measures spread/dispersion around the mean, not location.</p><p><strong>Step 2:</strong> For the variate <em>y</em> = (ax + b)/c, we can rewrite this as y = (a/c)·x + (b/c).</p><p><strong>Step 3:</strong> Applying the transformation rule: SD(y) = |a/c|·SD(x) = |a/c|·σ</p><p><strong>Step 4:</strong> The constant b contributes only to the mean shift and cancels in the deviation calculation. The division by c scales all deviations by the factor 1/c.</p><p><strong>Step 5:</strong> Therefore, SD = <strong>|a|σ/|c|</strong> or <strong>aσ/c</strong> (assuming a, c > 0)</p><p>∴ Answer: <strong>aσ/c</strong> (Option B)</p>
Correct Answer: B