Question

The median of the following data is 32.5, find the missing frequencies \(x\) and \(y\) :

Hint:

When dealing with missing frequencies, always form two equations: one from the total sum and one from the Median/Mean/Mode formula given. This ensures you can solve for both variables.

Answer 1

Step 1: Understanding the Concept:
The median involves cumulative frequencies. We use the sum of frequencies and the median formula to solve for two unknowns.
Step 3: Detailed Explanation:
Total frequency \(\sum f = 40\).
\(x + 5 + 9 + 12 + y + 3 + 2 = 40 \Rightarrow x + y + 31 = 40 \Rightarrow x + y = 9\) (Eq 1).
Median is 32.5, which falls in class 30 - 40.
Median Formula: \(M = l + \left( \frac{N/2 - cf}{f} \right) \times h\).
Here, \(l = 30, N/2 = 20, f = 12, h = 10\).
Cumulative frequency before median class (\(cf\)) = \(x + 5 + 9 = x + 14\).
\[ 32.5 = 30 + \left( \frac{20 - (x + 14)}{12} \right) \times 10 \]
\[ 2.5 = \left( \frac{6 - x}{12} \right) \times 10 \]
\[ 2.5 \times \frac{12}{10} = 6 - x \Rightarrow 3 = 6 - x \Rightarrow x = 3 \]
Substitute \(x = 3\) into Eq 1:
\[ 3 + y = 9 \Rightarrow y = 6 \]
Step 4: Final Answer:
The missing frequencies are \(x = 3\) and \(y = 6\).