A common mistake is using the cumulative frequency of the median class itself (\(cf=30\)) instead of the preceding class (\(cf=10\)). Always be careful to use the cf of the class *before* the median class.
Step 1: Understanding the Concept:
The median is the middle value of a dataset. For grouped frequency data, we first identify the median class and then use a specific formula to calculate the median value. Step 2: Key Formula or Approach:
The formula for the median of grouped data is:
\[ \text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h \]
where:
\(l\) = lower limit 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 size Step 3: Detailed Explanation:
First, we create a table with cumulative frequencies (cf).
Total frequency, \( N = \sum f = 50 \).
Next, we find the position of the median: \( \frac{N}{2} = \frac{50}{2} = 25 \).
The median class is the class interval whose cumulative frequency is just greater than or equal to 25. Looking at the cf column, the value just greater than 25 is 30, which corresponds to the class interval 20-30.
So, the median class is 20-30.
From this, we identify the values for the formula:
Lower limit of the median class, \(l = 20\).
Cumulative frequency of the preceding class, \(cf = 10\).
Frequency of the median class, \(f = 20\).
Class size, \(h = 10\) (e.g., 20 - 10 = 10).
Now, substitute these values into the median formula:
\[ \text{Median} = 20 + \left( \frac{25 - 10}{20} \right) \times 10 \]
\[ \text{Median} = 20 + \left( \frac{15}{20} \right) \times 10 \]
\[ \text{Median} = 20 + \frac{150}{20} \]
\[ \text{Median} = 20 + 7.5 \]
\[ \text{Median} = 27.5 \]
Step 4: Final Answer:
The median from the given data is 27.5.
