Find the median for the grouped data given below $:$
Marks
$50 - 60$
$60 - 70$
$70 - 80$
$80 - 90$
Students
$3$
$4$
$8$
$5$

53.75

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63.75

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73.75

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83.75

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Solution

The correct option is B $63.75$
Marks
Students(f)
Cumulative frequency(cf)
$50 - 60$
$3$
$3$
$60 - 70$
$4$
$3 + 4 = 7$
$70 - 80$
$8$
$7 + 8 = 15$
$80 - 90$
$5$
$15 + 5 = 20$

$n = Σ f = 20$
$\frac{n}{2} = \frac{20}{2} = 10$
Median $= l + ⎛ ⎜ ⎜ ⎝ \frac{\frac{n}{2} - c f}{f} ⎞ ⎟ ⎟ ⎠ \times h$
$n = Σ f =$ Total number of frequency $= 20$
$\frac{n}{2} = \frac{20}{2} = 10$
So, the median class is $70 - 80$
L $=$ Lower limit of the median class $= 70$
cf $=$ Cumulative frequency of the class preceding the median class $= 15$
f $=$ Frequency of the median class $= 8$
h $=$ class width $= 10$
Therefore, Median $= l + ⎛ ⎜ ⎜ ⎝ \frac{\frac{n}{2} - c f}{f} ⎞ ⎟ ⎟ ⎠ \times h$
$= 70 + ⎛ ⎜ ⎜ ⎜ ⎝ \frac{\frac{20}{2} - 15}{8} ⎞ ⎟ ⎟ ⎟ ⎠ \times 10$
$= 70 + (\frac{10 - 15}{8}) \times 10$
$= 70 - 6.25$
$= 63.75$
Hence, the median is $63.75$

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