Step 1: Understanding the Concept:
The median of a grouped distribution is calculated using the cumulative frequency and the median class formula. Step 2: Key Formula or Approach:
Median $= l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h$.
Also, the sum of all frequencies equals the total frequency ($n$). Step 3: Detailed Explanation:
1. Cumulative Frequencies ($cf$):
$x$, $x+5$, $x+14$, $x+26$, $x+26+y$, $x+29+y$, $x+31+y$.
2. Given Total Frequency $= 40$:
$x + y + 31 = 40 \implies x + y = 9$ (Equation 1).
3. Given Median $= 32.5$. This value lies in the class 30 - 40.
Median Class: 30 - 40 $\implies l = 30, f = 12, h = 10, cf = x+14, n/2 = 20$.
4. Apply Median Formula:
$32.5 = 30 + \left( \frac{20 - (x + 14)}{12} \right) \times 10$
$2.5 = \frac{6 - x}{12} \times 10 \implies \frac{2.5}{10} = \frac{6 - x}{12}$
$0.25 = \frac{6 - x}{12} \implies 3 = 6 - x \implies x = 3$.
5. Substitute $x=3$ in Equation 1:
$3 + y = 9 \implies y = 6$. Step 4: Final Answer:
The values are $x = 3$ and $y = 6$.
