Question

if the median of following data is 32 then find the value of x and y when N is 100​

Answer 1

Step-by-step explanation:

hope it'll help you have a wonderful day

Answer 2

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

The frequency distribution table is as

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c}\sf Class\: interval&\sf Frequency\: (f)&\sf \: cumulative \: frequency\\\frac{\qquad \qquad}{}&\frac{\qquad \qquad \qquad}{}\\\sf 0 - 10&\sf 10&\sf10\\\\\sf 10 - 20 &\sf x&\sf10 +x \\\\\sf 20-30 &\sf 25&\sf35 + x\\\\\sf 30 - 40&\sf 30&\sf65 + x\\\\\sf 40-50&\sf y&\sf65 + x + y\\\\\sf 50-60&\sf 10&\sf75 + x + y\\\frac{\qquad}{}&\frac{\qquad}{}\\\sf & \sf & \end{array}}\end{gathered}\end{gathered}\end{gathered}$

According to statement,

$\green{\bf :\longmapsto\: \sum \: f \: = \: 100}$

$\rm :\longmapsto\:75 + x + y = 100$

$\rm :\longmapsto\: x + y = 100 - 75$

$\rm :\longmapsto\: x + y = 25 - - - (1)$

We know, Median is evaluated as

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

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,

Median = 32

So,

Median class is 30-40

Therefore,

l = 30,

h = 10,

f = 30,

cf = cf of preceding class = 35 + x

N/2 = 50

By substituting all the given values in the formula,

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

$\dashrightarrow\sf 32= 30 + \Bigg \{10 \times \dfrac{ ( 50 - (35 + x))}{30} \Bigg \}$

$\dashrightarrow\sf 32 - 30 = \Bigg \{ \dfrac{ ( 50 - 35 - x)}{3} \Bigg \}$

$\dashrightarrow\sf 2 = \Bigg \{ \dfrac{ 15 - x}{3} \Bigg \}$

$\dashrightarrow\sf 6 = 15 - x$

$\dashrightarrow\sf x = 15 - 6$

$\dashrightarrow\sf x = 9$

On substituting the value of x, in equation (1), we get

$\dashrightarrow\sf y + 9 = 25$

$\dashrightarrow\sf y = 25 - 9$

$\dashrightarrow\sf y = 16$

Additional Information :-

Mode :- .

$\boxed{ \boxed{\sf{Mode = l + \bigg(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \bigg) \times h }}}$

Mean using Direct Method : -

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

Mean using Short Cut Method :-

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

Mean using Step Deviation Method :-

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