Calculate the mode of the following data.
Size of item $10 - 15$ $15 - 20$ $20 - 25$ $25 - 30$ $30 - 35$ $35 - 40$ $40 - 45$ $45 - 50$
No. of items $4$ $8$ $18$ $30$ $20$ $10$ $5$ $2$

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Solution

Size of the item $f$
$10 - 15$ $4$
$15 - 20$ $8$
$20 - 25$ $18$
$25 - 30$ $30$
$30 - 35$ $20$
$35 - 40$ $10$
$40 - 45$ $5$
$45 - 50$ $2$
Modal class is $25 - 30$ since it has the maximum frequency.
Lower limit of the modal class $l = 25$
Frequency of the modal class $f = 30$
Frequency of the preceding modal class $f_{1} = 18$
Frequency of the class reducing the modal class $f_{2} = 20$
Class width $c = 5$
Mode $= l + (\frac{f - f_{1}}{2 f - f_{1} - f_{2}}) \times c$
$= 25 + (\frac{30 - 18}{60 - 18 - 20}) \times 5 = 25 + \frac{12 \times 5}{22}$
$= 25 + \frac{60}{22} = 25 + 2.73 = 27.73$
Mode $= 27.73$

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Size (in cms):	10-15	15-20	20-25	25-30	30-35	35-40
No. of items:	2	8	20	35	20	15

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Class:	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$
Frequency:	$5$	$6$	$15$	$10$	$5$	$4$	$2$	$2$

Size	5−10	10−15	15−20	20−25	25−30	30−35	35−40	40−45	45−50
Frequency	6	10	18	30	15	12	10	6	4

CI	$0 - 5$	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$	$45 - 50$
$f$	$9$	$13$	$6$	$12$	$9$	$6$	$10$	$15$	$6$	$4$

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Size of item	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$	$45 - 50$
No. of items	$4$	$8$	$18$	$30$	$20$	$10$	$5$	$2$