To solve the problem based on the table provided:

Steps:

1. Median Calculation:

The total number of seedlings is $N = 47$. The median is located at the $\frac{N+1}{2}$-th value, i.e., at position $\frac{47+1}{2} = 24$-th value.

Using the cumulative frequency from the table:

• 10–13 cm: $f = 3$, cumulative frequency = 3
• 14–17 cm: $f = 6$, cumulative frequency = 9
• 18–21 cm: $f = 8$, cumulative frequency = 17
• 22–25 cm: $f = 12$, cumulative frequency = 29
• 26–29 cm: $f = 10$, cumulative frequency = 39
• 30–33 cm: $f = 6$, cumulative frequency = 45
• 34–37 cm: $f = 2$, cumulative frequency = 47

The 24th value falls in the interval 22–25 cm because the cumulative frequency reaches 29 in this interval. To calculate the exact median:

Using the formula: $\text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f_m}\right) \cdot h$ Where:

• $L = 22$ (lower boundary of the median class)
• $N/2 = 23.5$ (24th value is close to half)
• $CF = 17$ (cumulative frequency before the median class)
• $f_m = 12$ (frequency of the median class)
• $h = 4$ (class width for 22–25 cm)

Substitute: $\text{Median} = 22 + \left(\frac{23.5 - 17}{12}\right) \cdot 4$ $\text{Median} = 22 + \left(\frac{6.5}{12}\right) \cdot 4$ $\text{Median} = 22 + 2.17 \approx 24.17 \, \text{cm}$

2. Mode Calculation:

The mode is the value corresponding to the class with the highest frequency, which is $12$ in the interval 22–25 cm.

Using the formula for mode: $\text{Mode} = L + \left(\frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)}\right) \cdot h$ Where:

• $L = 22$ (lower boundary of modal class)
• $f_m = 12$ (frequency of modal class)
• $f_1 = 8$ (frequency of class before modal class)
• $f_2 = 10$ (frequency of class after modal class)
• $h = 4$ (class width)

Substitute: $\text{Mode} = 22 + \left(\frac{12 - 8}{(12 - 8) + (12 - 10)}\right) \cdot 4$ $\text{Mode} = 22 + \left(\frac{4}{4 + 2}\right) \cdot 4$ $\text{Mode} = 22 + \left(\frac{4}{6}\right) \cdot 4$ $\text{Mode} = 22 + 2.67 \approx 24.67 \, \text{cm}$

Final Answers:

• (a) Median = 24.17 cm
• (b) Mode = 24.67 cm

Let me know if you need further clarification or have more questions! Here are 5 related questions to deepen your understanding:

1. How would the calculations change if the total number of seedlings was different?
2. What does the cumulative frequency distribution tell us about the data?
3. Why is the median class determined by the 24th position?
4. What is the significance of the class width in both median and mode calculations?
5. How can the accuracy of the mode be improved in grouped data?

Tip: Always confirm that your cumulative frequencies sum to the total $N$ to avoid errors in median determination.