Question 4

We need to find the median length of the leaves.

Step 1 — Convert to Continuous Classes

The given classes are not continuous. We need to make them continuous. We subtract 0.5 from lower limits. We add 0.5 to upper limits. The new classes are now continuous.

| Length (in mm) (Continuous Classes) | Number of leaves (f) | | :---------------------------------- | :------------------- | | 117.5 - 126.5 | 3 | | 126.5 - 135.5 | 5 | | 135.5 - 144.5 | 9 | | 144.5 - 153.5 | 12 | | 153.5 - 162.5 | 5 | | 162.5 - 171.5 | 4 | | 171.5 - 180.5 | 2 |

Step 2 — Calculate Cumulative Frequencies

Let's add a cumulative frequency column. This helps us find the median class. The total number of leaves, $N$, is 40.

| Length (in mm) (Continuous Classes) | Number of leaves (f) | Cumulative Frequency (cf) | | :---------------------------------- | :------------------- | :------------------------ | | 117.5 - 126.5 | 3 | 3 | | 126.5 - 135.5 | 5 | 8 | | 135.5 - 144.5 | 9 | 17 | | 144.5 - 153.5 | 12 | 29 | | 153.5 - 162.5 | 5 | 34 | | 162.5 - 171.5 | 4 | 38 | | 171.5 - 180.5 | 2 | 40 |

Step 3 — Find Median Class

We find the value of $N/2$. $N/2 = 40/2$ $= 20$ We look for the cumulative frequency just greater than 20. This is 29. The class corresponding to 29 is the median class. The median class is 144.5 - 153.5.

Step 4 — Apply Median Formula

The median formula is: $\text{Median} = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$ Here, $l$ is the lower limit of the median class. $l = \textbf{144.5}$. $N/2 = \textbf{20}$. $cf$ is the cumulative frequency of the class before the median class. $cf = \textbf{17}$. $f$ is the frequency of the median class. $f = \textbf{12}$. $h$ is the class size. $h = 153.5 - 144.5$ $= \textbf{9}$ Now, we substitute these values into the formula. $\text{Median} = 144.5 + \left(\frac{20 - 17}{12}\right) \times 9$ $= 144.5 + \left(\frac{3}{12}\right) \times 9$ $= 144.5 + \left(\frac{1}{4}\right) \times 9$ $= 144.5 + 0.25 \times 9$ $= 144.5 + 2.25$

$\boxed{146.75 \text{ mm}}$

Answer

The median length of the leaves is 146.75 mm.