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Q4:

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 :

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.)

Solution :

Given: The frequency distribution of the lengths of 40 leaves measured to the nearest millimetre.

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

To Find: The median length of the leaves.

Step 1: Converting to Continuous Classes
The given classes are discontinuous (e.g., 126 and 127). To make them continuous, we subtract 0.5 from the lower limit and add 0.5 to the upper limit of each class interval.

Class Interval Frequency ($f$) Cumulative Frequency ($cf$)
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 $n = 40$ -

Step 2: Identifying the Median Class
The total number of observations is $n = 40$.
We calculate $\frac{n}{2} = \frac{40}{2} = 20$.
Looking at the cumulative frequency column, the first cumulative frequency greater than 20 is 29, which corresponds to the class interval $144.5 - 153.5$.
Thus, the median class is $144.5 - 153.5$.

Step 3: Applying the Median Formula
The formula for the median is:
$\text{Median} = l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h$
Where:
$l$ (lower limit of median class) = $144.5$
$n$ (total number of observations) = $40$
$cf$ (cumulative frequency of the class preceding the median class) = $17$
$f$ (frequency of the median class) = $12$
$h$ (class size) = $153.5 - 144.5 = 9$

Step 4: Calculation
$\text{Median} = 144.5 + \left( \frac{20 - 17}{12} \right) \times 9$
$\text{Median} = 144.5 + \left( \frac{3}{12} \right) \times 9$
$\text{Median} = 144.5 + \left( \frac{1}{4} \right) \times 9$
$\text{Median} = 144.5 + 2.25$
$\text{Median} = 146.75$

Final Answer: The median length of the leaves is 146.75 mm.


More Questions from Class 10 Mathematics Statistics EXERCISE 13.3


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