The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table : Length (in mm) Number of leaves 118 - 126 3 127 - 135 5 136 - 144 9 145 - 153 12 154 - 162 5 163 - 171 4 172 - 180 2 Find the median length of the leaves. (Hint : The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 - 126.5, 126.5 - 135.5, . . ., 171.5 - 180.5.) 200
Step-by-Step Solution
Key Concept: For grouped (continuous) data, the median is found using the formula \(\displaystyle \text{Median}=L+\frac{\frac{N}{2}-c_f}{f}\times h\), where \(L\) is the lower limit of the median class, \(c_f\) is the cumulative frequency before the median class, \(f\) is the frequency of the median class and \(h\) is the class width.
1. Convert to continuous classes\
Original class \(118-126\) becomes \(117.5-126.5\), \(127-135\) becomes \(126.5-135.5\), and so on. The class width \(h\) is \(126.5-117.5 = 9\) mm.\
2. Tabulate frequencies and cumulative frequencies\
| Class (mm) | Frequency \(f\) | Cumulative \(c_f\) |
|------------|----------------|-------------------|
| 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 |
Total \(N = 40\).\
3. Locate the median class\
Median position = \(\frac{N}{2}=20\) (or \(\frac{N+1}{2}=20.5\)). The cumulative frequency just before 20 is 17 (after the third class) and after the fourth class it becomes 29. Hence the median class is \(144.5‑153.5\).\
4. Apply the median formula\
\[\text{Median}=L+\frac{\frac{N}{2}-c_f}{f}\times h\]\
Here, \(L = 144.5\) mm, \(c_f = 17\), \(f = 12\), \(h = 9\) mm.\
\[\text{Median}=144.5+\frac{20-17}{12}\times 9\]
\[\text{Median}=144.5+\frac{3}{12}\times 9\]
\[\text{Median}=144.5+0.25\times 9\]
\[\text{Median}=144.5+2.25\]
\[\text{Median}=146.75\ \text{mm}\]
5. State the answer\
The median length of the leaves is \(146.75\) mm (≈ \(146.8\) mm to one decimal place).
Correct Answer: 146.75 mm (≈ 146.8 mm)