Math, asked by namratapatel6399, 4 months ago

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Class 10th
Chapter - Statistics
Board - CBSE​

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Answers

Answered by mathdude500
3

Evaluation of Median

1. \:  \bf \: Median  \: formula : \\

 \boxed{ { \purple{\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

⠀⠀⠀⠀</p><p>\begin{gathered}\boxed{\begin{array}{ccc}\sf Class\: interval&amp;\sf F_i&amp;\sf c.frequency\\\frac{\quad \qquad\qquad}{}&amp;\frac{\quad \qquad \qquad}{}&amp;\\\sf 0-10&amp;\sf 2&amp;\sf 2&amp;\\\\\  \sf 10-20 &amp;\sf 3&amp; \sf 5\\\\\sf 20-30 &amp;\sf 9&amp;\sf 14&amp;\\\\\sf 30-40&amp;\sf 2&amp;\sf 16&amp;\\\\\sf 40-50 &amp;\sf 6 &amp;\sf 22&amp;\\\frac{\qquad \qquad \qquad}{}&amp;\frac{\qquad \qquad \qquad}{}&amp;\frac{\qquad\qquad\qquad}{}&amp;\frac{\qquad \qquad\qquad}{}&amp;\frac{\qquad \qquad\qquad}{}\\\sf Total&amp; \sf \sum f = 22&amp;&amp;&amp; \sf\end{array}}\end{gathered}

we know that N = 22 and N/2 = 11

The cumulative frequency just greater than 11 is 14 and the corresponding class is 20-30

According to the question,

Median class is 20-30

So,

l = 20, h = 10, f = 9, cf = cf of preceding class = 5 and N/2 = 11

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= 20 + \Bigg \{10 \times \dfrac{ ( 11 - 5)}{9} \Bigg \}

\dashrightarrow\sf M= 20 + \bigg \{ 10 \times \dfrac{(6)}{9} \bigg \}

\dashrightarrow\sf M= 20 + (6.67)

</p><p>\dashrightarrow\sf M= 26.67 \: (approx)

 \boxed{ \red{ \bf \: Thus, Median = 26.67}}

Evaluation of Mode

\begin{gathered}\Large{\bold{\pink{\underline{Formula \:  Used \::}}}}  \end{gathered}

Formula for mode:

2. \: \boxed{ \boxed{\sf{mode = l + \bigg(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \bigg) \times h }}}

Where,

l is lower limit of modal class.

\sf{f_1} \: is  \: frequency  \: of  \: modal  \: class

\sf{f_0} \:  is \:  frequency  \: of  \: class \:  preceding \:  modal  \: class

\sf{f_2} \:  is \:  frequency \:  of  \: class \:  succeeding \:  modal \:  class,  \: and

 \sf \: h \:  is  \: class \:  height.

\begin{gathered} \begin{array}{|c|c|} \bf{x_i} &amp; \bf{f_i} \\ 0 - 10 &amp; 2  \\10 - 20 &amp; 3 \\20 - 30 &amp; 9 \\30 - 40 &amp; 2 \\40 - 50 &amp; 6 \end{array}\end{gathered}

Here,

According to the question,

Modal class is 20 - 30.

 \bf \: f_0 =3 , f_1 = 9, f_2 =2 , l = 20, h = 10

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

{{\bf{Mode = 20 + \bigg(\dfrac{9 - 3}{2 \times 9 - 3 - 2} \bigg) \times 10 }}}

{{\bf{Mode = 20 + \bigg(\dfrac{6}{18 - 5} \bigg) \times 10 }}}

{{\bf{Mode = 20 + \bigg(\dfrac{6}{13} \bigg) \times 10 }}}

{{\bf{Mode = 20 + \bigg(\dfrac{60}{13} \bigg) }}}

{{\bf{Mode = 20 \:  +  \: 4.61 }}}

 \boxed{ \red{{{\bf{Mode = 24.61 }}}}}

Evaluation of Mean

\begin{gathered}\Large{\bold{\pink{\underline{Formula \:  Used \::}}}}  \end{gathered}

\boxed{ \sf \overline{x} = A + \dfrac{\sum f_iu_i}{\sum f_i}  \times h}

⠀⠀⠀⠀</p><p>\begin{gathered}\boxed{\begin{array}{ccccc}\sf Class\: interval&amp;\sf F_i&amp;\sf Mid\:Value&amp;\sf u_i = \dfrac{x_i + 25}{10}&amp;\sf f_i u_i\\\frac{\quad \qquad\qquad}{}&amp;\frac{\quad \qquad \qquad}{}&amp;\\\sf 0-10&amp;\sf 2&amp;\sf 5&amp;\sf -2&amp;\sf -4\\\\\sf 10-20 &amp;\sf 3&amp;\sf 15&amp;\sf -1 &amp;\sf -3 \\\\\sf 20-30 &amp;\sf 9&amp;\sf A = 25&amp;\sf 0&amp;\sf 0\\\\\sf 30-40&amp;\sf 2&amp;\sf 35&amp;\sf 1 &amp;\sf 2\\\\\sf 40-50 &amp;\sf 6 &amp;\sf 45&amp;\sf 2&amp;\sf 12\\\frac{\qquad \qquad \qquad}{}&amp;\frac{\qquad \qquad \qquad}{}&amp;\frac{\qquad\qquad\qquad}{}&amp;\frac{\qquad \qquad\qquad}{}&amp;\frac{\qquad \qquad\qquad}{}\\\sf Total&amp; \sf \sum f = 22&amp;&amp;&amp; \sf 7\end{array}}\end{gathered}

Therefore, mean is given by

:  \implies  \bf \: { \sf \overline{x} = A + \dfrac{\sum f_iu_i}{\sum f_i}  \times h}

:  \implies  \bf \: { \sf \overline{x} = 25 + \dfrac{7}{22}  \times 10}

:  \implies  \bf \: { \sf \overline{x} = 25 +3.18}

 \implies \boxed{ \red{  \bf \: { \bf \overline{x} = 28.18}}}

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