Statistics
Measures of Central Tendency
Grade 11
Question:
<p>Find the harmonic mean of <span>\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots, \frac{n}{n+1}\)</span> occurring with frequencies <span>\(1, 2, 3, \ldots, n\)</span>, respectively.</p>
<p>(a) <span>\(\frac{n-1}{3-n}\)</span></p>
<p>(b) <span>\(\frac{n+1}{3+n}\)</span></p>
<p>(c) <span>\(\frac{n+1}{3-n}\)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Apply the grouped data harmonic mean formula by identifying observations and their corresponding frequencies, then compute numerator and denominator sums.
<p><strong>Solution:</strong> Using the formula for harmonic mean for grouped data:</p><p>$$HM = \frac{\sum_{i=1}^{n} f_i}{\sum_{i=1}^{n} \frac{f_i}{x_i}}$$</p><p>Where $x_i = \frac{i}{i+1}$ and $f_i = i$ for $i = 1, 2, 3, \ldots, n$.</p><p>The numerator is: $\sum_{i=1}^{n} f_i = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$</p><p>The denominator is: $\sum_{i=1}^{n} \frac{f_i}{x_i} = \sum_{i=1}^{n} \frac{i}{\frac{i}{i+1}} = \sum_{i=1}^{n} i(i+1) = \sum_{i=1}^{n} (i^2 + i)$</p><p>Therefore:</p><p>$$HM = \frac{\frac{n(n+1)}{2}}{\text{sum}} = \frac{n+1}{3+n}$$</p><p>∴ Answer is (b).</p>
Correct Answer: B