Question

Find the median of the following data:

Hint:

To find the median of grouped data, use the cumulative frequency table and the median formula.

Answer 1

To find the median, we first calculate the cumulative frequency. The cumulative frequency is the running total of the frequencies:

The total number of data points \( N = 60 \).
The median class is the class interval where the cumulative frequency is greater than or equal to \( \frac{N}{2} = 30 \).
From the cumulative frequency table, the median class is \( 20 - 30 \), as the cumulative frequency reaches 40 here, which is greater than 30.
Now, to find the median, we use the formula:
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h, \]
where:

- \( L \) is the lower boundary of the median class (\( 20 \)),
- \( F \) is the cumulative frequency before the median class (\( 10 \)),
- \( f \) is the frequency of the median class (\( 30 \)),
- \( h \) is the class width (\( 10 \)).
Substitute the values:
\[ \text{Median} = 20 + \left( \frac{30 - 10}{30} \right) \times 10. \] Simplify:
\[ \text{Median} = 20 + \left( \frac{20}{30} \right) \times 10 = 20 + \frac{200}{30} = 20 + 6.67 = 26.67. \]
Conclusion:
The median of the given data is \( 26.67 \).