Question

The frequency distribution table shows the number of mango trees in a grove and their yield of mangoes. Find the median of the data:
\[ \begin{array}{|c|c|} \hline \text{No. of Mangoes} & \text{No. of Trees (f)} \\ \hline 50 - 100 & 33 \\ 100 - 150 & 30 \\ 150 - 200 & 90 \\ 200 - 250 & 80 \\ 250 - 300 & 17 \\ \hline \end{array} \]

Hint:

To find the median of grouped data, locate the median class where cumulative frequency ≥ \(N/2\), then apply \[ \text{Median} = L + \frac{(N/2 - c.f.)}{f} \times h \]

Answer 1

Step 1: Find the cumulative frequency (c.f.).
\[ \begin{array}{|c|c|c|} \hline \text{Class Interval} & \text{Frequency (f)} & \text{Cumulative Frequency (c.f.)} \\ \hline 50 - 100 & 33 & 33 \\ 100 - 150 & 30 & 63 \\ 150 - 200 & 90 & 153 \\ 200 - 250 & 80 & 233 \\ 250 - 300 & 17 & 250 \\ \hline \end{array} \] Step 2: Find total frequency.
\[ N = 250 \] Step 3: Find median class.
\[ \frac{N}{2} = \frac{250}{2} = 125 \] The c.f. just greater than 125 is 153. Hence, the median class is \(150 - 200\). Step 4: Apply the median formula.
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - c.f.}{f} \right) \times h \] Where, \(L = 150\), \(N = 250\), \(c.f. = 63\), \(f = 90\), \(h = 50\) \[ \text{Median} = 150 + \left( \frac{125 - 63}{90} \right) \times 50 \] \[ \text{Median} = 150 + \left( \frac{62}{90} \times 50 \right) \] \[ \text{Median} = 150 + 34.44 = 184.44 \] Step 5: Conclusion.
The median number of mangoes produced per tree is approximately \(184.4\).
Final Answer: \[ \boxed{\text{Median = 184.4}} \]