Question

find the mode of the following distribution monthly wages200-220,220-240,240-260,260-280,280-300,300-320
no of workes7,15,20,20,10,2​

Answer 1

Step-by-step explanation:

I hope it will be help full for you

Answer 2

$\large\underline{\sf{Solution-}}$

Here, the largest frequency 20 repeated twice.

• So, its a bi-modal series.

Hence, to find the mode of series, we use Empirical Formula.

So, we first evaluate Median.

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c}\sf Class\: interval&\sf Frequency\: (f)&\sf \: cumulative \: frequency\\\frac{\qquad\qquad}{}&\frac{\qquad \qquad}{}\\\sf 200 - 220&\sf 15&\sf15\\\\\sf 220 - 240 &\sf 7&\sf22\\\\\sf 240-260 &\sf 20&\sf42\\\\\sf 260 - 280&\sf 20&\sf62\\\\\sf 280-300&\sf 10&\sf72\\\\\sf 300-320&\sf 2&\sf74\\\frac{\qquad}{}&\frac{\qquad}{}\\\sf & \sf & \end{array}}\end{gathered}\end{gathered}\end{gathered}$

Median formula -

$\rm :\longmapsto\:\boxed{ \sf M= l + \Bigg \{h \times \dfrac{ \bigg( \dfrac{N}{2} - cf \bigg)}{f} \Bigg \}}$

Here,

• l denotes lower limit of median class

• h denotes width of median class

• f denotes frequency of median class

• cf denotes cumulative frequency of the class preceding the median class

• N denotes sum of frequency

According to the question,

• Here, N = 74

So,

• N/2 = 37

• median class is 240 - 260

so,

• l = 240,

• h = 20,

• f = 20,

• cf = cf of preceding class = 23

By substituting all the given values in the formula,

$\dashrightarrow\sf M= l + \Bigg \{h \times \dfrac{ \bigg( \dfrac{N}{2} - cf \bigg)}{f} \Bigg \}$

$\dashrightarrow\sf M= 240 + \Bigg \{20 \times \dfrac{ ( 37 - 22)}{20} \Bigg \}$

$\dashrightarrow\bf M= 240 + 15 = 255 - - (1)$

Now, we evaluate mean of the given data.

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c|c|c}\sf Class\: interval&\sf Frequency\: (f_i)&\sf \: midvalue \: (x_i)&\sf \: u_i&\sf \: f_iu_i\\\frac{\qquad \qquad}{}&\frac{\qquad \qquad}{}\\\sf 200 - 220&\sf 15&\sf210&\sf - 2&\sf - 30\\\\\sf 220 - 240 &\sf 7&\sf230&\sf - 1&\sf - 7\\\\\sf 240-260 &\sf 20 &\sf250 -A &\sf0&\sf0\\\\\sf 260 - 280&\sf 20&\sf270&\sf1&\sf20\\\\\sf 280-300&\sf 10&\sf290&\sf2&\sf20\\\\\sf 300-320&\sf 2&\sf310&\sf3&\sf6\\\frac{\qquad}{}&\frac{\qquad}{}\\\sf & \sf & \end{array}}\end{gathered}\end{gathered}\end{gathered}$

Here,

$\: \: \: \: \: \: \: \: \bull \: \sf \: A \: = \: 250$

$\: \: \: \: \: \: \: \: \bull \: \sf \: h \: = \: 20$

$\: \: \: \: \: \: \: \: \bull \: \sf \: \sum \: \: f_i \: = \: 74$

$\: \: \: \: \: \: \: \: \bull \: \sf \: \sum \: f_iu_i \: = \: 9$

Mean is given by

$\dashrightarrow\sf Mean = \: A \: + \: \dfrac{ \sum f_i u_i}{ \sum f_i} \times h$

$\dashrightarrow\sf Mean = \: 250 + \dfrac{9}{74} \times 20$

$\dashrightarrow\sf Mean = \: 250 + \dfrac{37}{90}$

$\dashrightarrow\sf Mean = \: 250 + 2.43$

$\dashrightarrow\sf Mean =252.43$

Hence,

Mode is evaluated using Empirical Formula as

$\rm :\longmapsto\:Mode =3Median - 2Mean$

$\rm :\implies\:Mode = 3 \times 255 - 2 \times 252.43$

$\bf :\implies\:Mode = 765 - 504.86 = 260.14$

Additional Information :-

Mean using direct Formula

$\dashrightarrow\sf Mean = \dfrac{ \sum f_i x_i}{ \sum f_i}$

Mean using Short Cut Method

$\dashrightarrow\sf Mean = \: A \: + \: \dfrac{ \sum f_i d_i}{ \sum f_i}$