Question

A survey conducted on 20 household in a locality by a group of students resulted in the following frequency table for the number of family in a household.

family size 1-3, 3-5, 5-7, 7-9, 9-11
no of
families. 7. 8. 2. 2. 1

find the mode of the data​

Answer 1

Solution :-

$\begin{tabular}{c | l} Class Interval & Frequency \\ \cline{1-2} 1 - 3 & 7 \\ 3 - 5 & 8 \\ 5 - 7 & 2 \\ 7 - 9 & 2 \\ 9 - 11 & 1 \end{tabular}$

From table

The class interval 3 - 5 has the highest frequecy among all class intervals. So 3 - 5 is the Modal class

Lower boundary of the modal class (l) = 3

Frequency of the modal class (f1) = 8

Frequency of the class preceeding the modal class (f0) = 7

Frequency of the class succeeding the modal class (f2) = 2

Class size (h) = Lower boundary of one class - Lower boundery of preeceeding the class = 3 - 1 = 2

$\bf Mode = l + \bigg( \dfrac{f_1 - f_0 }{2f_ 1 - f_0 - f_2} \bigg) \times h$

$\\ \\ \sf = 3 + \bigg( \dfrac{8 - 7}{2(8) - 7 -2 } \bigg) \times 2 \\ \\ \sf = 3 + \bigg( \dfrac{1}{28) - 7 - 2 } \bigg) \times 2 \\ \\ \sf = 3 + \bigg( \dfrac{ 1}{16 - 9 } \bigg) \times 2 \\ \\ \sf = 3 + \bigg( \dfrac{1}{7} \bigg) \times 2 \\ \\ \sf = 3 + \dfrac{2}{7} \\ \\ \sf = \dfrac{21 + 2}{7} \\ \\ \sf = \dfrac{23}{7} \\ \\ \sf = 3.28$

Mode = 3.28

Therefore the mode of the data is 3.28.

Answer 2

Given :

• l = 3
• f1 = 8
• fo = 7
• f2 = 2
• h = 2 (3 - 1)

Find :

The mode.

Solution :

We know that..

Mode = $l \: + \: \bigg (\dfrac{ f_{1} \: - \: f_{o} }{2f_{1} \: - \: f_{o} \: - \: f_{2}} \bigg) \: \times \: h$

Here..

• l = lower limit of the modal class
• f1 = frequency of the modal class
• f2 = frequency of the class succeeding the modal class
• fo = frequency of the class preceding the modal class
• h = lower boundary of one class - lower boundary of preceding class.

Now,

Put the known values in above formula

=> Mode = $3\: + \: \bigg (\dfrac{ 8 \: - \: 7 }{2(8)\: - \: 7 \: - \: 2} \bigg) \: \times \: 2$

=> $3\: + \: \bigg (\dfrac{ 1 }{16\: - \: 9} \bigg) \: \times \: 2$

=> $3\: + \: \bigg (\dfrac{ 1 }{7} \bigg) \: \times \: 2$

=> $3 \: + \: \dfrac{2}{7}$

=> $\dfrac{21 \: + \: 2}{7}$

=> $\dfrac{23}{7}$

=> $3.28$

∴ Mode is 3.28