The lengths 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 Number of leaves fi- 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-

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Standard XI

Statistics

Median of Discrete Series

The lengths o...

Question

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

$\begin{matrix} L e n g t h (i n m m) & N u m b e r o f l e a v e s f_{i} 118 - 126 & 3 127 - 135 & 5 136 - 144 & 9 145 - 153 & 12 154 - 162 & 5 163 - 171 & 4 172 - 180 & 2 \end{matrix}$

Find the median length of the leaves.

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Solution

The given data does not have continuous class intervals. We can observe that difference between two class intervals is 1. So, we have to add and subtract $\frac{1}{2} = 0.5$ from upper-class limits and lower class limits respectively.

Now, continuous class intervals with respective cumulative frequencies can be represented as below:

$\begin{matrix} L e n g t h (i n m m) & N u m b e r o f l e a v e s f_{i} & C u m u l a t i v e f r e q u e n c y 117.5 - 126.5 & 3 & 3 126.5 - 135.5 & 5 & 3 + 5 = 8 135.5 - 144.5 & 9 & 8 + 9 = 17 144.5 - 153.5 & 12 & 17 + 12 = 29 153.5 - 162.5 & 5 & 29 + 5 = 34 162.5 - 171.5 & 4 & 34 + 4 = 38 171.5 - 180.5 & 2 & 38 + 2 = 40 \end{matrix}$

From the table, we observe that cumulative frequency just greater then $\frac{n}{2} (i . e . \frac{40}{2} = 20)$ is 29, belonging to class interval 144.5 - 153.5.

Median class = 144.5 - 153.5

Lower limit $l$ of median class = 144.5

Class size $h$ = 9

Frequency $f$ of median class = 12

Cumulative frequency $c f$ of class preceding median class = 17, $M e d i a n = l + (\frac{\frac{n}{2} - c_{f}}{f}) \times h = 144.5 + (\frac{20 - 17}{12}) \times 9 = 144.5 + \frac{9}{4} = 146.75$

So, median length of leaves is 146.75 mm.

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Similar questions

Q. The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:

\begin{matrix} 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 \end{matrix}

Convert given table into a continuous group frequency distribution.

Question 4 (i)
The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:
$\begin{matrix} Length(in mm) & Number of leaves 118 - 126 & 3 127 - 135 & 5 136 - 144 & 9 145 - 153 & 12 154 - 162 & 5 163 - 171 & 4 172 - 180 & 2 \end{matrix}$
(i) Draw a histogram to represent the given data.

Question 4 (ii)
The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:
$\begin{matrix} Length (in mm) & Number of leaves 118 - 126 & 3 127 - 135 & 5 136 - 144 & 9 145 - 153 & 12 154 - 162 & 5 163 - 171 & 4 172 - 180 & 2 \end{matrix}$
(ii) Is there any other suitable graphical representation for the same data?

Question 4 (iii)
The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:
$\begin{matrix} Length (in mm) & Number of leaves 118 - 126 & 3 127 - 135 & 5 136 - 144 & 9 145 - 153 & 12 154 - 162 & 5 163 - 171 & 4 172 - 180 & 2 \end{matrix}$
(iii) Is it correct to conclude that the maximum number of leaves are 153 mm long? Why?

Q. 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
$118 - 126$ $127 - 135$ $136 - 144$ $145 - 153$ $154 - 162$ $163 - 171$ $172 - 180$	$3$ $5$ $9$ $12$ $5$ $4$ $2$

Find the median length of the leaves.

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The lengths 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)Number of leaves fi118−1263127−1355136−1449145−15312154−1625163−1714172−1802 Find the median length of the leaves.