Question:
<p>The average of \(n\) numbers \(x_1, x_2, x_3, \ldots, x_n\) is \(M\). If \(x_n\) is replaced by \(x'\), then new average is</p>
<p>\(M - x_n + x'\)</p>
<p>\(\dfrac{nM - x_n + x'}{n}\)</p>
<p>\(\dfrac{(n-1)M + x'}{n}\)</p>
<p>\(\dfrac{M - x_n + x'}{n}\)</p>
Step-by-Step Solution
Key Concept: When one data value changes, the new mean shifts by the change in that value divided by the total count. The new average equals the old average plus the net change per observation.
<p><strong>Step 1:</strong> The sum of all n numbers is nM (since average M = Sum/n).</p><p><strong>Step 2:</strong> When x_n is replaced by x', the new sum becomes: nM - x_n + x'</p><p><strong>Step 3:</strong> The new average is the new sum divided by n:</p><p>New Average = (nM - x_n + x')/n = M + (x' - x_n)/n</p><p><strong>Step 4:</strong> This can also be written as: <strong>[nM - x_n + x']/n</strong> or <strong>M - (x_n - x')/n</strong></p><p>∴ Answer: <strong>M + (x' - x_n)/n</strong> or equivalent form <strong>(nM - x_n + x')/n</strong></p>
Correct Answer: B