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Statistics
NCERT Exemplar
CBSE
Grade 10
Question:
If the median of the distribution given below is $28.5$, find the values of $x$ and $y$. Class: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, Total Frequency: 5, x, 20, 15, y, 5, 60
Step-by-Step Solution
Key Concept: Total frequency $N = 60 \Rightarrow 45 + x + y = 60 \Rightarrow x + y = 15$. Median is $28.5$, so median class is $20-30$ ($l=20, f=20, cf=5+x, h=10$). Apply median formula.
Total frequency: $5 + x + 20 + 15 + y + 5 = 60 \Rightarrow 45 + x + y = 60 \Rightarrow x + y = 15$ -- (eq 1). [1.0 Mark] Given Median $= 28.5$, which lies in class $20-30$. So median class is $20-30$. Here $l=20, f=20, cf = 5 + x, h=10, N/2 = 30$. [1.0 Mark] $\text{Median} = l + \left(\dfrac{N/2 - cf}{f}\right)h \Rightarrow 28.5 = 20 + \left(\dfrac{30 - (5 + x)}{20}\right) \times 10$. [1.0 Mark] $8.5 = \dfrac{25 - x}{2} \Rightarrow 17 = 25 - x \Rightarrow x = 8$. [1.0 Mark] Substitute $x = 8$ into eq 1: $8 + y = 15 \Rightarrow y = 7$. Values: $x = 8$ and $y = 7$. [1.0 Mark]
--- 🎯 Official CBSE Marking Scheme: Equation $x + y = 15$: 1.0 Mark Identifying median class $20-30$ and parameters: 1.0 Mark Applying median formula $28.5 = 20 + (25-x)/2$: 1.0 Mark Solving for $x = 8$: 1.0 Mark Solving for $y = 7$: 1.0 Mark
Correct Answer:
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