Question

Consider the following frequency distribution: 
 
If the sum of all frequencies is 584 and median is 45, then \(|\alpha - \beta|\) is equal to __________ 
 

Hint:

In median problems for grouped data, always determine the median class first. It is the interval in which the given median value falls. This fixes $L, f, CF$ and $h$ for the equation.

Answer 1

Correct Answer: 164

Step 1: Understanding the Concept:
The median of a grouped frequency distribution is found using the median class formula.
We use the given sum of frequencies and the median value to establish two linear equations in \(\alpha\) and \(\beta\).
Step 2: Key Formula or Approach:
1. \(\sum f = N\).
2. \(Median = L + \left( \frac{N/2 - CF}{f} \right) \times h\).
Step 3: Detailed Explanation:
1. Total frequency equation:
\(\alpha + 110 + 54 + 30 + \beta = 584 \implies \alpha + \beta = 390\).

2. Median calculation:
Median \(= 45\). This lies in the class interval 40-50.
So, Median Class is 40-50.
Lower limit \(L = 40\), class width \(h = 10\), frequency \(f = 30\).
Total \(N = 584 \implies N/2 = 292\).
Cumulative Frequency (\(CF\)) before class 40-50 is \(\alpha + 110 + 54 = \alpha + 164\).
Substitute into formula:
\[ 45 = 40 + \left( \frac{292 - (\alpha + 164)}{30} \right) \times 10 \]
\[ 5 = \frac{128 - \alpha}{3} \implies 15 = 128 - \alpha \implies \alpha = 113 \]

3. Find \(\beta\) and the difference:
\(\beta = 390 - 113 = 277\).
\(|\alpha - \beta| = |113 - 277| = 164\).
Step 4: Final Answer:
The absolute difference is 164.