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Statistics
RD Sharma
CBSE
Grade 10

Question:

The median of the following frequency distribution of 100 students is $32$. Find the missing frequencies $f_1$ and $f_2$:
Marks: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60
Students: 10, f1, 25, 30, f2, 10

Step-by-Step Solution

Key Concept: Total students $= 75 + f_1 + f_2 = 100 \Rightarrow f_1 + f_2 = 25$.<br>Median $= 32 \Rightarrow$ Median class is $30-40$ ($l = 30, f = 30, cf = 35 + f_1, h = 10$).<br>$32 = 30 + \left(\dfrac{50 - (35 + f_1)}{30}\right) \times 10 \Rightarrow 2 = \dfrac{15 - f_1}{3} \Rightarrow 6 = 15 - f_1 \Rightarrow f_1 = 9$.<br>Since $f_1 + f_2 = 25 \Rightarrow f_2 = 16$. Missing frequencies are $f_1 = 9, f_2 = 16$.
Sum of frequencies: $75 + f_1 + f_2 = 100 \Rightarrow f_1 + f_2 = 25$. [1.0 Mark]
Median $= 32 \Rightarrow$ Median class $30-40$ ($l=30, f=30, cf=35+f_1, h=10$). [1.0 Mark]
$32 = 30 + \left(\dfrac{50 - 35 - f_1}{30}\right) \times 10 \Rightarrow 2 = \dfrac{15 - f_1}{3}$. [1.5 Marks]
$6 = 15 - f_1 \Rightarrow f_1 = 9$. [1.0 Mark]
$f_2 = 25 - 9 = 16$. Missing frequencies are $f_1 = 9, f_2 = 16$. [0.5 Mark]

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🎯 Official CBSE Marking Scheme:
Forming linear relation $f_1 + f_2 = 25$: 1.0 Mark
Identifying median class $30-40$ and $cf = 35+f_1$: 1.0 Mark
Solving $f_1 = 9$: 1.5 Marks
Solving $f_2 = 16$: 1.5 Marks

Correct Answer:
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