Question

The length of 40 leaves of a plant are measured correct to the nearest millimeter, and the data obtained is represented in the following table:

Length (in mm):	118−126	127−135	136−144	145−153	154−162	163−171	172−180
No. of leaves	3	5	9	12	5	4	2

Find the mean length of leaf.

Answer 1

Calculation for mean.

Length (in mm)	Mid-Values(xi)	Number of Leaves(fi )	fi xi
117.5–126.5	122	3	366
126.5–135.5	131	5	655
135.5–144.5	140	9	1260
144.5–153.5	149	12	1788
153.5–162.5	158	5	790
162.5–171.5	167	4	668
171.5–180.5	176	2	353
		N = 40	$\sum _{}{f}_i{x}_i=$5880

Mean length of the leaf = $\frac{1}{N}\sum _{}^{}{f}_i{x}_i=\frac{1}{40}\times 5880=147$

Calculation for median.

The given series is in inclusive form. Converting it to exclusive form and preparing the cumulative frequency table, we have

Length (in mm)	Number of Leaves(fi )	Cumulative Frequency (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
	N = 40

Now, we have

So,

Now, the cumulative frequency just greater than 20 is 29 and the corresponding class is 144.5–153.5.

Therefore, 144.5–153.5 is the median class.

Here,

We know that

$$\begin{gathered} =144.5+\left(\frac{20-17}{12}\right)\times 9 \\ =144.5+\frac{27}{12} \\ =144.5+2.25 \\ =146.75 \end{gathered}$$

Hence, the median length of leaf is 146.75 mm.

Disclaimer: If the question asks for the mean length of the leaf, then the answer is 147 mm whereas if the question asks fro the median length of the leaf, then the answer is 146.75 mm, which is same as the answer given in the book.