In an examination taken by $400$ students, the scores were as shown in the following distribution table:
Marks $1 - 10$ $11 - 21$ $21 - 30$ $31 - 40$ $41 - 50$
$51 - 60$ $61 - 70$ $71 - 80$ $81 - 90$ $91 - 100$
Frequency $8$ $14$ $32$ $56$ $102$ $80$ $54$ $30$ $16$ $8$
Find the mode scores?

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Solution

$M a r k s$ $F r e q u e n c y$
$0.5 - 10.5$ $8$
$10.5 - 20.5$ $14$
$20.5 - 30.5$ $32$
$30.5 - 40.5$ $56$
$40.5 - 50.5$ $102$
$50.5 - 60.5$ $80$
$60.5 - 70.5$ $54$
$70.5 - 80.5$ $30$
$80.5 - 90.5$ $16$
$90.5 - 100.5$ $8$
$T o t a l$ $400$
Here, modal class $= 40.5 - 50.5$
So, $l = 40.5, f_{0} = 56, f_{1} = 102, f_{2} = 80$ and $h = 10$
$M o d e = l + \frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}} \times h$

$= 40.5 + \frac{102 - 56}{2 \times 102 - 56 - 80} \times 10$

$= 40.5 + 0.68$
$= 41.18$

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Marks	$1 - 10$	$11 - 21$	$21 - 30$	$31 - 40$	$41 - 50$	$51 - 60$	$61 - 70$	$71 - 80$	$81 - 90$	$91 - 100$
Frequency	$8$	$14$	$32$	$56$	$102$	$80$	$54$	$30$	$16$	$8$