<p>The standard deviation of 25 numbers is 40. If each of the numbers is increased by 5, then the new standard deviation will be</p>
Step-by-Step Solution
Key Concept: Standard deviation measures spread around the mean. Adding a constant to all data points shifts the mean but preserves the deviation of each point from the new mean, leaving standard deviation unchanged.
<p><strong>Step 1:</strong> Recall the definition of standard deviation. If data set is {x₁, x₂, ..., x₂₅} with mean μ, then σ = √[Σ(xᵢ - μ)²/n]</p><p><strong>Step 2:</strong> When each number is increased by 5, new data set becomes {x₁+5, x₂+5, ..., x₂₅+5}. The new mean becomes μ + 5.</p><p><strong>Step 3:</strong> For each new observation: (xᵢ + 5) - (μ + 5) = xᵢ - μ. The deviations from the new mean remain identical to the deviations from the original mean.</p><p><strong>Step 4:</strong> Therefore, σ_new = √[Σ(xᵢ + 5 - (μ + 5))²/25] = √[Σ(xᵢ - μ)²/25] = 40</p><p>∴ Answer: The new standard deviation is <strong>40</strong> (unchanged)</p>
Correct Answer: A