Calculate mode of the following distribution:

Marks 10−19 20−29 30−39 40−49 50−59 60−69 70−79 80−89 90−99

Number of Students 29 87 181 247 263 133 40 9 2

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Solution

Note that the given distribution is in the form of inclusive class intervals. For the calculation of mode, first the class intervals must be converted into exclusive form using the following formula.
$Value of Adjustment = \frac{Lower limit of one class - Upper limit of the preceeding class}{2}$ Value of lower limit of one class − Value of upper limit of the preceeding class2
The value of adjustment as calculated is then added to the upper limit of each class and subtracted from the lower limit of each class. In this manner, we get the following distribution.

Class Interval Exclusive Class Interval Frequency
(f)

10 − 19
20 − 29
30 − 39
40 − 49
50 − 59
60 − 69
70 − 79
80 − 89
90 − 99 9.5 − 19.5
19.5 − 29.5
29.5 − 39.5
39.5 − 49.5
49.5 − 59.5
59.5 − 69.5
69.5 − 79.5
79.5 − 89.5
89.5 − 99.5 29
87
181
247 = f₀
263 = f₁
133 = f₂
40
9
2

By inspection, we can say that the modal class is (49.5 – 59.5) as it has the highest frequency of 263.
$Mode (Z) = l_{1} + \frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}} \times i or, Z = 49.5 + \frac{263 - 247}{2 (263) - 247 - 133} \times 10 or, Z = 49.5 + \frac{16}{526 - 380} \times 10 or, Z = 49.5 + \frac{16}{146} \times 10 or, Z = 49.5 + 1.09 = 50.59 Hence, mode is 50.59$

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Similar questions

X	10−19	20−29	30−39	40−49	50−59	60−69	70−79	80−89
No. Persons	32	42	40	56	20	6	2	2

Marks (more than)	0-9	10-19	20-29	30-39	40-49	50-59
No of students	2	5	15	25	10	3

Marks	20−29	30−39	40−49	50−59	60−69	70−79
No. of Students	4	10	12	9	7	3

Following table shows the marks of 120 students at an examination. Draw an ogive for the table: estimate Inter-quartile range.

Class-intervals	10−29	30−39	40−49	50−59	59−60	60−69
Frequencies	12	19	20	21	15	13

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Class Interval	Exclusive Class Interval	Frequency (f)
10 − 19 20 − 29 30 − 39 40 − 49 50 − 59 60 − 69 70 − 79 80 − 89 90 − 99	9.5 − 19.5 19.5 − 29.5 29.5 − 39.5 39.5 − 49.5 49.5 − 59.5 59.5 − 69.5 69.5 − 79.5 79.5 − 89.5 89.5 − 99.5	29 87 181 247 = f₀ 263 = f₁ 133 = f₂ 40 9 2