Question

UNIT I Determine the median wage graphically from the following data: Wages No. of workers Wages No. of workers 700-800 4 800-900 6 900-1000 10 1000-1100 16 1100-1200 12 1200-1300 7 1300-1400 3​

Answer 1

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

The frequency distribution table is as follow

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c}\sf Class\: interval&\sf Frequency\: (f)&\sf \: cumulative \: frequency\\\frac{\qquad \qquad}{}&\frac{\qquad \qquad}{}\\\sf 700 - 800&\sf 4&\sf4\\\\\sf 800 - 900 &\sf 6&\sf10\\\\\sf 900-1000 &\sf 10&\sf20\\\\\sf 1000 - 1100&\sf 16&\sf36\\\\\sf 1100-1200&\sf 12&\sf48\\\\\sf 1200-1300&\sf 7&\sf55\\\\\sf 1300-1400&\sf 3&\sf58\\\frac{\qquad}{}&\frac{\qquad}{}\\\sf & \sf & \end{array}}\end{gathered}\end{gathered}\end{gathered}$

Now, we know that

➢ Median of continuous series is given by

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

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 Calculations,

➢ Median class is 1000-1100

So,

l = 1000,

h = 100,

f = 16,

cf = cf of preceding class = 20

N/2 = 29

By substituting all the given values in the formula,

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

$\dashrightarrow\sf M= 1000+ \Bigg \{100 \times \dfrac{ \bigg( 29 - 20 \bigg)}{16} \Bigg \}$

$\dashrightarrow\sf M= 1000+ \Bigg \{100 \times \dfrac{ 9}{16} \Bigg \}$

$\dashrightarrow\sf M= 1000+ \Bigg \{25 \times \dfrac{ 9}{4} \Bigg \}$

$\dashrightarrow\sf M= 1000+ \Bigg \{\dfrac{225}{4} \Bigg \}$

$\dashrightarrow\sf M= 1000+ 56.25$

$\dashrightarrow\sf M= 1056.25$

Hence,

$\\ \rm\implies \:\boxed{\tt{ \: \: Median \: = \: 1056.25 \: \: }}$

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MORE TO KNOW

1. Mean using Direct Method

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

2. Mean using Short Cut Method

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

3. Mean using Step Deviation Method

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

Answer 2

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

The frequency distribution table is as follow

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c}\sf Class\: interval&\sf Frequency\: (f)&\sf \: cumulative \: frequency\\\frac{\qquad \qquad}{}&\frac{\qquad \qquad}{}\\\sf 700 - 800&\sf 4&\sf4\\\\\sf 800 - 900 &\sf 6&\sf10\\\\\sf 900-1000 &\sf 10&\sf20\\\\\sf 1000 - 1100&\sf 16&\sf36\\\\\sf 1100-1200&\sf 12&\sf48\\\\\sf 1200-1300&\sf 7&\sf55\\\\\sf 1300-1400&\sf 3&\sf58\\\frac{\qquad}{}&\frac{\qquad}{}\\\sf & \sf & \end{array}}\end{gathered}\end{gathered}\end{gathered}$

Now, we know that

➢ Median of continuous series is given by

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

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 Calculations,

➢ Median class is 1000-1100

So,

l = 1000,

h = 100,

f = 16,

cf = cf of preceding class = 20

N/2 = 29

By substituting all the given values in the formula,

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

$\dashrightarrow\sf M= 1000+ \Bigg \{100 \times \dfrac{ \bigg( 29 - 20 \bigg)}{16} \Bigg \}$

$\dashrightarrow\sf M= 1000+ \Bigg \{100 \times \dfrac{ 9}{16} \Bigg \}$

$\dashrightarrow\sf M= 1000+ \Bigg \{25 \times \dfrac{ 9}{4} \Bigg \}$

$\dashrightarrow\sf M= 1000+ \Bigg \{\dfrac{225}{4} \Bigg \}$

$\dashrightarrow\sf M= 1000+ 56.25$

$\dashrightarrow\sf M= 1056.25$

Hence,

$\\ \rm\implies \:\boxed{\tt{ \: \: Median \: = \: 1056.25 \: \: }}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

MORE TO KNOW

1. Mean using Direct Method

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

2. Mean using Short Cut Method

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

3. Mean using Step Deviation Method

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