Find the value of x for the following distribution whose mean is 31.87

1.	Find the value of x for the following distribution whose mean is 31.87 .
Answer» QUESTION find the value of x for the following distribution whose MEAN is 31.87 . $\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{cccc}\sf x_i &\sf f_i\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf 12&\sf \ 8 \\\\\sf 20 &\sf \ 16 \\\\\sf 27&\sf \ 48 \\\\\sf 33&\sf 90\\\\\sf x&\sf 30 \\\\\sf 54&\sf 8 \\\\\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{\bf{}}\end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$ ANSWER $\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{cccc}\sf x_i &\sf f_i&\sf f_ix_i \\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 12&\sf \ 8 &\sf 96 \\\\\sf 20 &\sf \ 16 &\sf 320 \\\\\sf 27&\sf \ 48&\sf 1296 \\\\\sf 33&\sf 90&\sf 2970\\\\\sf x&\sf 30 &\sf 30x \\\\\sf 54&\sf 8 &\sf 432 \\\\\sf Total &\sf \Sigma{f_i}=200&\sf \Sigma{f_ix_i}=5114 + 30x\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{\bf{}}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$ Now , $\sf mean \: = \dfrac{\sf \Sigma{f_ix_i}}{ \sf \Sigma{f_i}}$ $\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 31.87 = \frac{5114 + 30x}{200} \\\end{gathered}\end{gathered} \end{gathered}$ $\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 6374 = 5114 + 30x \\\end{gathered}\end{gathered} \end{gathered}$ $\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 30x=6374 - 5114 \\\end{gathered}\end{gathered} \end{gathered}$ $\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 30x=1260 \\\end{gathered}\end{gathered} \end{gathered}$ $\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf x = \dfrac{1260}{30} = 42 \\\end{gathered}\end{gathered} \end{gathered}$ $\begin{gathered} \begin{gathered}\begin{gathered} \Longrightarrow \underline{ \boxed{\displaystyle \sf hence \: x = 42 }} \\ \\\end{gathered}\end{gathered}\end{gathered}$ ____________________ additional information mean of tabular data If $\sf x_1,x_2,x_3,....,x_n$ are n observations with frequencies $\sf f_1,f_2,f_3,....,f_n$ respectively , then the mean x of these observation is GIVEN by $\sf x = \dfrac{f_1x_1 +f_2x_2 + f_3x_3 +... + f_nx_n} {f_1 +f_2 + f_3+... + f_n} = \dfrac{\Sigma{f_ix_i} }{\Sigma{f_i}}$ where $\Sigma$ ( CALLED sigma ) is the GREEK letter representing summation.

Find the value of x for the following distribution whose mean is 31.87 .

Answer»

QUESTION

find the value of x for the following distribution whose MEAN is 31.87 .

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{cccc}\sf x_i &\sf f_i\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf 12&\sf \ 8 \\\\\sf 20 &\sf \ 16 \\\\\sf 27&\sf \ 48 \\\\\sf 33&\sf 90\\\\\sf x&\sf 30 \\\\\sf 54&\sf 8 \\\\\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{\bf{}}\end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

ANSWER

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{cccc}\sf x_i &\sf f_i&\sf f_ix_i \\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 12&\sf \ 8 &\sf 96 \\\\\sf 20 &\sf \ 16 &\sf 320 \\\\\sf 27&\sf \ 48&\sf 1296 \\\\\sf 33&\sf 90&\sf 2970\\\\\sf x&\sf 30 &\sf 30x \\\\\sf 54&\sf 8 &\sf 432 \\\\\sf Total &\sf \Sigma{f_i}=200&\sf \Sigma{f_ix_i}=5114 + 30x\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{\bf{}}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

Now ,

$\sf mean \: = \dfrac{\sf \Sigma{f_ix_i}}{ \sf \Sigma{f_i}}$

$\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 31.87 = \frac{5114 + 30x}{200} \\\end{gathered}\end{gathered} \end{gathered}$

$\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 6374 = 5114 + 30x \\\end{gathered}\end{gathered} \end{gathered}$

$\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 30x=6374 - 5114 \\\end{gathered}\end{gathered} \end{gathered}$

$\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 30x=1260 \\\end{gathered}\end{gathered} \end{gathered}$

$\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf x = \dfrac{1260}{30} = 42 \\\end{gathered}\end{gathered} \end{gathered}$

$\begin{gathered} \begin{gathered}\begin{gathered} \Longrightarrow \underline{ \boxed{\displaystyle \sf hence \: x = 42 }} \\ \\\end{gathered}\end{gathered}\end{gathered}$

____________________

additional information

mean of tabular data

If $\sf x_1,x_2,x_3,....,x_n$ are n observations with frequencies $\sf f_1,f_2,f_3,....,f_n$ respectively , then the mean x of these observation is GIVEN by

$\sf x = \dfrac{f_1x_1 +f_2x_2 + f_3x_3 +... + f_nx_n} {f_1 +f_2 + f_3+... + f_n} = \dfrac{\Sigma{f_ix_i} }{\Sigma{f_i}}$

where $\Sigma$ ( CALLED sigma ) is the GREEK letter representing summation.