Statistics
Standard Deviation
Grade 11
Question:
<p>Let \(x_1, x_2, x_3, x_4, x_5\) be the observations with mean \(m\) and standard deviation \(s\). The standard deviation of the observations \(kx_1, kx_2, kx_3, kx_4, kx_5\) is</p>
<p>\(k+s\)</p>
<p>\(\dfrac{s}{k}\)</p>
<p>\(ks\)</p>
<p>\(s\)</p>
Step-by-Step Solution
Key Concept: When all observations are multiplied by a constant k, the standard deviation scales by the absolute value of that constant (σ_new = |k|·σ_old), while the mean scales identically (μ_new = k·μ_old). This follows because standard deviation measures spread, which scales proportionally with the magnitude of k.
<p><strong>Step 1:</strong> Recall that standard deviation σ = √[Σ(xᵢ - μ)²/n]</p><p><strong>Step 2:</strong> For observations kx₁, kx₂, kx₃, kx₄, kx₅, the mean becomes km (since mean scales linearly).</p><p><strong>Step 3:</strong> The new standard deviation is: σ_new = √[Σ(kxᵢ - km)²/n] = √[Σk²(xᵢ - m)²/n] = √[k²·Σ(xᵢ - m)²/n] = |k|·√[Σ(xᵢ - m)²/n] = |k|·s</p><p><strong>Step 4:</strong> Therefore, the standard deviation of the new observations is <strong>ks</strong> (or |k|s if k could be negative, but typically in JEE context this is written as ks).</p><p>∴ Answer: C</p>
Correct Answer: C