The lengths of 40 leaves of an plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table. Find the median length of the leaves.

Question :

Length (in mm)	118 – 126	127 – 135	136 – 144	145 – 153	154 – 162	163 – 171	172 – 180
Number of Leaves	3	5	9	12	5	4	2

Solution :

Here the frequency table is given in the inclusive form. So, we first convert it into exclusive form by subtracting and adding \(h\over 2\) to the lower and upper limits respectively of each class, where h denotes the difference of lower limit of a class and the upper limit of the previous class.

Converting the given table into exclusive form and preparing the cumulative frequency table, we get

Length (in mm)	Number of leaves (frequency)	Cumulative frequency
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

We have : n = 40 So, \(n\over 2\) = 20

The cumulative frequency just greater than \(n\over 2\) is 29 and the corresponding class is (144.5 – 153.5).

So, it is the median class.

Here, l = 144.5, h = 9, f = 12 and cf = 17

Substituting these values in the formula

Median = l + (\({n\over 2} – cf\over f\) \(\times\) h,

Median = 144.5 + \(3\over 12\) \(\times\) 9 = 144.5 + 2.25 = 146.75 mm