Question

If the median of the following frequency table is 28.5, find the values of $x$ and $y$ where the sum of frequencies is 80. 

Hint:

To identify the median class, find where the cumulative frequency just exceeds $\dfrac{N}{2}$. Substitute carefully in the median formula.

Answer 1

Step 1: Given that total frequency = 80. 
\[ 5 + x + 20 + 15 + y + 5 = 80 \Rightarrow x + y = 35 \quad \text{...(1)} \] Step 2: Median formula. 
\[ \text{Median} = l + \left(\frac{\frac{N}{2} - c.f.}{f}\right) \times h \] Step 3: Find median class. 
Median = 28.5, $\Rightarrow$ lies in 20–30 class. So, \[ l = 20, \; h = 10, \; f = 20, \; N = 80, \; \frac{N}{2} = 40 \] Cumulative frequency before median class: $5 + x = (5 + x)$. 
Step 4: Substitute values. 
\[ 28.5 = 20 + \left(\frac{40 - (5 + x)}{20}\right) \times 10 \] 
Step 5: Simplify. 
\[ 28.5 - 20 = \frac{(35 - x)}{2} \] \[ 8.5 = \frac{35 - x}{2} \Rightarrow 17 = 35 - x \Rightarrow x = 18 \] 
Step 6: Find $y$. 
From (1): $x + y = 35 \Rightarrow 18 + y = 35 \Rightarrow y = 17$. 
Step 7: Conclusion. 
\[ \boxed{x = 18, \; y = 17} \]