Question

Write the median class of the data.

Answer 1

Median Class

The cumulative frequency is calculated as follows:

\(\text{Length (in mm)}\)	\(\text{Number of Leaves}\)	\(\text{Cumulative Frequency (CF)}\)
70 – 80	3	3
80 – 90	5	8
90 – 100	9	17
100 – 110	12	29
110 – 120	5	34
120 – 130	4	38
130 – 140	2	40

The total number of leaves is 40, so the median will lie at the 20th position. From the cumulative frequency, the 20th leaf lies in the class 100–110, so the median class is 100–110.

Answer 2

Leaves of length equal to or more than 10 cm (100 mm):

The relevant classes are:

• 100–110: 12 leaves
• 110–120: 5 leaves
• 120–130: 4 leaves
• 130–140: 2 leaves

Total = 12 + 5 + 4 + 2 = 23 leaves of length ≥ 10 cm.

Answer 3

(a) To find the median, use the formula for grouped data:

\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h \]

Where:

• \( L = 100 \) (lower boundary of the median class)
• \( N = 40 \) (total number of leaves)
• \( F = 17 \) (cumulative frequency before median class)
• \( f = 12 \) (frequency of the median class)
• \( h = 10 \) (class width)

\[ \text{Median} = 100 + \left( \frac{20 - 17}{12} \right) \times 10 = 100 + 2.5 = 102.5 \, \text{mm} \]

Thus, the Median is \( 102.5 \, \text{mm} \).

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(b) The modal class is the class with the highest frequency, which is 100-110 with 12 leaves. To find the mode, use the formula:

\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \]

Where:

• \( L = 100 \) (lower boundary of the modal class)
• \( f_1 = 12 \) (frequency of the modal class)
• \( f_0 = 9 \) (frequency of the class before the modal class)
• \( f_2 = 5 \) (frequency of the class after the modal class)
• \( h = 10 \) (class width)

\[ \text{Mode} = 100 + \left( \frac{12 - 9}{2 \times 12 - 9 - 5} \right) \times 10 = 100 + \left( \frac{3}{10} \right) \times 10 = 100 + 3 = 103 \, \text{mm} \]

Thus, the Mode is \( 103 \, \text{mm} \).