Statistics
Standard Deviation
Grade None
Question:
<p>Let \(a, b, c, d\) and \(e\) be the observations with mean \(m\) and standard deviation \(s\). The standard deviation of the observations \(a+k, b+k, c+k, d+k, e+k\) is</p>
<p>\(ks\)</p>
<p>\(s\)</p>
<p>\(s+k\)</p>
<p>\(\dfrac{s}{k}\)</p>
Step-by-Step Solution
Key Concept: Adding a constant to all observations shifts the mean but does not change the spread or dispersion of the data. Standard deviation measures spread from the mean, which remains unchanged when a constant is added to all values.
<p><strong>Step 1:</strong> Original observations: a, b, c, d, e with mean = m and standard deviation = s</p><p><strong>Step 2:</strong> New observations: (a+k), (b+k), (c+k), (d+k), (e+k)</p><p><strong>Step 3:</strong> New mean = (a+k + b+k + c+k + d+k + e+k)/5 = (a+b+c+d+e)/5 + k = m + k</p><p><strong>Step 4:</strong> Standard deviation formula: σ = √[Σ(xᵢ - mean)²/n]</p><p><strong>Step 5:</strong> For new data: σ_new = √[Σ((xᵢ+k) - (m+k))²/5] = √[Σ(xᵢ - m)²/5] = √[Σ(xᵢ - m)²/n] = s</p><p><strong>Step 6:</strong> The deviations from the mean remain identical; only the mean shifts. Therefore, standard deviation is unchanged.</p><p>∴ Answer: B (Standard deviation remains s)</p>
Correct Answer: B