Question

$\huge{\underline{\underline{\bf Question}}}$

If the median of the distribution given above is 28.5, find the values of x and y.

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Answer 1

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

Frequency distribution table is as below

$\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c}\sf Class\: interval&\sf Frequency\: (f)&\sf \: cumulative \: frequency\\\frac{ \qquad \qquad}{}&\frac{\qquad \qquad}{}\\\sf 0 - 10&\sf 5&\sf5\\\\\sf 10 - 20 &\sf x&\sf5 + x\\\\\sf 20-30 &\sf 20&\sf25 + x\\\\\sf 30 - 40&\sf 15&\sf40 + x\\\\\sf 40-50&\sf y&\sf40 + x + y\\\ \\ \sf 50 - 60&\sf 5&\sf45 + x + y\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf & \sf \sum f = 60& \end{array}}\end{gathered}\end{gathered}\end{gathered}$

Now,

Given that

$\sf \: \sum \: f \: = \: 60$

$\rm :\implies\:45 + x + y = 60$

$\bf :\implies\:x + y = 15 - - - (1)$

We know,

The Formula of Median is

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

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,

• Median is 28.5

So,

• median class is 20-30

• l = 20,

• h = 10,

• f = 20,

• cf = cf of preceding class = 5 + x

• N/2 = 30

By substituting all the given values in the formula,

$\rm :\longmapsto\:28.5 = 20 + \dfrac{30 - (5 + x)}{20} \times 10$

$\rm :\longmapsto\:28.5 - 20 = \dfrac{30 - 5 - x}{2}$

$\rm :\longmapsto\:8.5 = \dfrac{25 - x}{2}$

$\rm :\longmapsto\:17 = 25 - x$

$\bf\implies \:x \: = \: 8 - - - (2)$

On substituting, the value of 'x = 8' in equation (1) we get

$\rm :\longmapsto\:8 + y = 15$

$\bf\implies \:y \: = \: 7$

Additional Information :-

$\begin{gathered} \dashrightarrow\sf Mode = x_{k} + \bigg \{h \times \dfrac{(f_{k} - f_{k - 1})}{(2f_{k} - _{k - 1} - f_{k + 1}} \bigg \} \\ \end{gathered}$

Mean using direct method :-

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

Mean using Assumed mean method :-

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

And

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