Question

The lengths of 40 leaves in a plant are measured correctly to the nearest millimeter, and the data obtained is represented as in the following table: Find the median length of leaves.

Length in mm	Number of leaves
$118-126$	$3$
$127-135$	$5$
$136-144$	$9$
$145-153$	12
$154-162$	$5$
$163-171$	$4$
$172-180$	$2$

Answer 1

Step 1. Calculate the cumulative frequency:

• 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..

We know that,

Median = $\mathrm{l}+\left[\frac{(\frac{\mathrm{n}}{2}-\mathrm{cf})}{\mathrm{f}}\right]\times \mathrm{h}$

• Class size, h
• number of observations, n
• Lower limit of median class, l
• Frequency of median class, f
• Cumulative frequency of class preceding median class, cf

Let's construct a continuous class data

Length in mm	Class interval	Number of leaves	Frequency
$118-126$	$117.5-126.5$	$3$	$3$
$127-135$	$126.5-135.5$	$5$	$3+5=8$
$136-144$	$135.5-144.5$	$9$	$8+9=17\mathrm{F}$
$145-153$	$144.5-153.5$	$12\mathrm{f}$	$17+12=29$
$154-162$	$153.5-162.5$	$5$	$29+5=34$
$163-171$	$162.5-171.5$	$4$	$34+4=38$
$172-180$	$171.5-180.5$	$2$	$38+2=40\mathrm{n}$

From the table, it can be observed that

$\begin{array}{rcl}\mathrm{n}& =& 40\\ & \Rightarrow & \frac{\mathrm{n}}{2}=20\end{array}$

Cumulative frequency (cf) just greater than 20 is 29, belonging to the class $144.5-153.5$

Therefore, the median class $=144.5-153.5$

Class size lower$\mathrm{h}=9$

Lower limit of median class, $\mathrm{l}=144.5$

Frequency of median class,$\mathrm{f}=12$

Cumulative frequency of class preceding median class,$\mathrm{cf}=17$

Step 2. calculate the median.

$$\begin{gathered} \mathrm{Median} \\ =\mathrm{l}+\left[\frac{(\frac{\mathrm{n}}{2}-\mathrm{cf})}{\mathrm{f}}\right]\times \mathrm{h} \end{gathered}$$

$\text{= 144.5 + [}\frac{(20-17)}{12}]\times 9$

$=144.5+\left(\frac{3}{12}\right)\times 9$

$=144.5+\frac{9}{4}$

$=144.5+2.25$

$=146.75$

Therefore, the median length of leaves is $146.75\mathrm{mm}.$