Statistics
Standard Deviation
Grade 11
Question:
<p>If <span>\(\sum_{i=1}^{n}(x_i - a) = n\)</span> and <span>\(\sum_{i=1}^{n}(x_i - a)^2 = na\)</span>, where <span>\(n, a > 1\)</span>, then the standard deviation of <span>\(n\)</span> observations <span>\(x_1, x_2, \ldots, x_n\)</span> is</p>
<p>(a) <span>\(a - 1\)</span></p>
<p>(b) <span>\(n\sqrt{a - 1}\)</span></p>
<p>(c) <span>\(n(a - 1)\)</span></p>
<p>(d) <span>\(\sqrt{a - 1}\)</span></p>
Step-by-Step Solution
Key Concept: Use the relationship between sum of deviations and mean, then apply the variance formula using given sums.
<p><strong>Step 1:</strong> From the given condition <span>\(\sum_{i=1}^{n}(x_i - a) = n\)</span>, we get the mean:</p><p><span>\(\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i = a + 1\)</span></p><p><strong>Step 2:</strong> Using the formula for variance:</p><p><span>\(\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - a)^2 - \left(\frac{1}{n}\sum_{i=1}^{n}(x_i - a)\right)^2\)</span></p><p><span>\(= \frac{na}{n} - \left(\frac{n}{n}\right)^2 = a - 1\)</span></p><p><strong>Step 3:</strong> Therefore, standard deviation <span>\(\sigma = \sqrt{a - 1}\)</span></p><p>∴ Answer is (d).</p>
Correct Answer: D