Question

In grouped frequency distribution, the formula to find median is:

• A. \( l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h \)
• B. \( l - \left( \frac{\frac{n}{2} - cf}{f} \right) \)
• C. \( l + \left( \frac{\frac{n}{2} + cf}{f} \right) \times h \)
• D. \( l \pm \left( \frac{\frac{n}{2} + cf}{2f} \right) \)

Hint:

To remember the grouped median formula easily: \[ \text{Median} = l + \left( \frac{\frac{n}{2} - cf}{f} \right)\times h \] Always remember:
• Start from the lower boundary \(l\)
• Subtract the previous cumulative frequency \(cf\)
• Multiply by class width \(h\)

Answer 1

The Correct Option is A

Concept: The median for grouped data is the value that divides the entire distribution into two equal parts. Half of the observations lie below the median and the remaining half lie above it. When data is arranged in the form of class intervals with corresponding frequencies, the median cannot usually be identified directly. Therefore, we use a standard interpolation formula to calculate it. The formula for median in grouped frequency distribution is: \[ \text{Median} = l + \left( \frac{\frac{n}{2} - cf}{f} \right)\times h \] where:
• \(l\) = lower boundary of the median class
• \(n\) = total frequency
• \(cf\) = cumulative frequency of the class preceding the median class
• \(f\) = frequency of the median class
• \(h\) = class width or class size

Step 1: Understanding the meaning of the median class.
The median class is the class interval whose cumulative frequency first becomes greater than or equal to: \[ \frac{n}{2} \] This class contains the median value. Once the median class is identified, we apply the interpolation formula to estimate the exact median within that class interval.

Step 2: Understanding the structure of the formula.
The formula begins from the lower boundary \(l\) of the median class. Then we calculate how far inside the class interval the median lies. The quantity: \[ \left( \frac{n}{2} - cf \right) \] represents the number of observations needed within the median class to reach the middle observation. Dividing by the class frequency \(f\) gives the fractional position inside the class. Multiplying by class width \(h\) converts this fraction into an actual distance within the interval. Thus: \[ \text{Median} = l + \left( \frac{\frac{n}{2} - cf}{f} \right)\times h \]

Step 3: Comparing the formula with the given options.
Option (1): \[ l + \left( \frac{\frac{n}{2} - cf}{f} \right)\times h \] This exactly matches the standard median formula. Hence, it is correct. Option (2): Incorrect because:
• the multiplication by class width \(h\) is missing
• the negative sign is incorrect Option (3): Incorrect because: \[ \frac{n}{2} + cf \] is mathematically wrong in the median formula. Option (4): Incorrect because:
• the formula structure is incorrect
• denominator \(2f\) is wrong
• sign convention is incorrect Final Conclusion: The correct formula for median in grouped frequency distribution is: \[ \boxed{ l + \left( \frac{\frac{n}{2} - cf}{f} \right)\times h } \] Hence, the correct answer is option (1).