Find the median length of the leaves from the given data. Find the median marks obtained by 80 students. Find the median wage of the workers from the provided wage data.

Steps to Solve

1. Calculate Cumulative Frequencies for Leaves

   For the leaves dataset, calculate the cumulative frequencies from the given frequency table:

   • Length $118-126$: 3
   • Length $127-135$: $3 + 5 = 8$
   • Length $136-144$: $8 + 9 = 17$
   • Length $145-153$: $17 + 12 = 29$
   • Length $154-162$: $29 + 5 = 34$
   • Length $163-171$: $34 + 4 = 38$
   • Length $172-180$: $38 + 2 = 40$

   The cumulative frequency table will look like this:

   Length (in mm)   | No. of leaves | Cumulative Frequency
   ------------------|----------------|----------------------
   118 - 126        | 3              | 3
   127 - 135        | 5              | 8
   136 - 144        | 9              | 17
   145 - 153        | 12             | 29
   154 - 162        | 5              | 34
   163 - 171        | 4              | 38
   172 - 180        | 2              | 40
   

2. Find the Median Class for Leaves

   With 40 leaves, the median position is given by: $$ \text{Median position} = \frac{40 + 1}{2} = 20.5 $$

   The cumulative frequency that first exceeds 20.5 is 29, corresponding to the class $145-153$. Thus, the median class is $145-153$.

3. Calculate the Median Length of Leaves Using the Formula

   The median is calculated using the formula: $$ \text{Median} = L + \left(\frac{N/2 - CF}{f}\right) \cdot c $$

   Where:

   • $L = 145$ (lower boundary of the median class)
   • $N = 40$ (total number of observations)
   • $CF = 17$ (cumulative frequency before the median class)
   • $f = 12$ (frequency of the median class)
   • $c = 8$ (class width)

   Plug in the values: $$ \text{Median} = 145 + \left(\frac{20 - 17}{12}\right) \cdot 8 $$

4. Evaluate the Median Length of Leaves

   Calculate: $$ \text{Median} = 145 + \left(\frac{3}{12}\right) \cdot 8 $$ $$ = 145 + 2 $$ $$ = 147 $$

5. Calculate Cumulative Frequencies for Marks

   For the marks dataset, calculate the cumulative frequencies:

   • Below 10: 3
   • Below 20: $3 + 12 = 15$
   • Below 30: $15 + 27 = 42$
   • Below 40: $42 + 57 = 99$
   • Below 50: $99 + 75 = 174$
   • Below 60: $174 + 80 = 254$

   The cumulative frequency table:

   Marks           | No. of students | Cumulative Frequency
   ----------------|------------------|----------------------
   Below 10       | 3                | 3
   Below 20       | 12               | 15
   Below 30       | 27               | 42
   Below 40       | 57               | 99
   Below 50       | 75               | 174
   Below 60       | 80               | 254
   

6. Find the Median Class for Marks

   With 80 students, the median position: $$ \text{Median position} = \frac{80 + 1}{2} = 40.5 $$

   The cumulative frequency that first exceeds 40.5 is 42, corresponding to the class "Below 30." Thus, the median class is "Below 30".

7. Calculate the Median Marks Using the Formula

   Using the same median formula: $$ \text{Median} = L + \left(\frac{N/2 - CF}{f}\right) \cdot c $$

   Where:

   • $L = 30$ (lower boundary of the median class)
   • $N = 80$ (total number of observations)
   • $CF = 15$ (cumulative frequency before the median class)
   • $f = 27$ (frequency of the median class)
   • $c = 10$ (class width)

   Plug in the values: $$ \text{Median} = 30 + \left(\frac{40 - 15}{27}\right) \cdot 10 $$

8. Evaluate the Median Marks

   Calculate: $$ \text{Median} = 30 + \left(\frac{25}{27}\right) \cdot 10 $$ $$ = 30 + 9.26 $$ $$ \approx 39.26 $$

9. Calculate Cumulative Frequencies for Wages

   For the wages dataset, calculate cumulative frequencies:

   • More than 150: 0
   • More than 140: $0 + 10 = 10$
   • More than 130: $10 + 29 = 39$
   • More than 120: $39 + 60 = 99$

   The cumulative frequency table:

   Wages (in ₹)    | No. of workers | Cumulative Frequency
   -----------------|-----------------|----------------------
   More than 150   | 0                | 0
   More than 140   | 10               | 10
   More than 130   | 29               | 39
   More than 120   | 60               | 99
   

10. Find the Median Class for Wages

With 100 workers, the median position: $$ \text{Median position} = \frac{100 + 1}{2} = 50.5 $$

The cumulative frequency that first exceeds 50.5 is 99, corresponding to the class "More than 120." Thus, the median class is "More than 120".

11. Calculate the Median Wages Using the Formula

Using the median formula: $$ \text{Median} = L + \left(\frac{N/2 - CF}{f}\right) \cdot c $$

Where:

• $L = 120$ (lower boundary of the median class)
• $N = 100$ (total number of observations)
• $CF = 39$ (cumulative frequency before the median class)
• $f = 60$ (frequency of the median class)
• $c = 10$ (class width)

Plug in the values: $$ \text{Median} = 120 + \left(\frac{50 - 39}{60}\right) \cdot 10 $$

12. Evaluate the Median Wages

Calculate: $$ \text{Median} = 120 + \left(\frac{11}{60}\right) \cdot 10 $$ $$ = 120 + 1.83 $$ $$ \approx 121.83 $$