Statistics
Mean of Frequency Distribution
Grade 11
Question:
<p>If the mean of the distribution is 2.6, then the value of \(y\) is</p><table border='1'><tr><th>Variate \(x\)</th><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><th>Frequency \(f\) of \(x\)</th><td>4</td><td>5</td><td>\(y\)</td><td>1</td><td>2</td></tr></table>
<p>24</p>
<p>13</p>
<p>8</p>
<p>3</p>
Step-by-Step Solution
Key Concept: Use the formula for mean: mean = Σ(x·f)/Σf. Set up an equation with the given mean of 2.6 and solve for the unknown frequency y.
<p><strong>Step 1:</strong> Write the mean formula:</p><p>Mean = Σ(x·f)/Σf = 2.6</p><p><strong>Step 2:</strong> Calculate numerator Σ(x·f):</p><p>= 1(4) + 2(5) + 3(y) + 4(1) + 5(2)</p><p>= 4 + 10 + 3y + 4 + 10 = 28 + 3y</p><p><strong>Step 3:</strong> Calculate denominator Σf:</p><p>= 4 + 5 + y + 1 + 2 = 12 + y</p><p><strong>Step 4:</strong> Set up the equation:</p><p>(28 + 3y)/(12 + y) = 2.6</p><p><strong>Step 5:</strong> Cross multiply:</p><p>28 + 3y = 2.6(12 + y)</p><p>28 + 3y = 31.2 + 2.6y</p><p>3y - 2.6y = 31.2 - 28</p><p>0.4y = 3.2</p><p>y = 8</p><p><strong>Verification:</strong> (28 + 24)/(12 + 8) = 52/20 = 2.6 ✓</p><p>∴ Answer: D (y = 8)</p>
Correct Answer: D