Question

the sum of deviation of a certain number of observation measured from 4 is 72 and the sum of observation of the sane value from 7 is -3 . find the number of observation and their mean. ​

Answer 1

$\textbf{Given:}$

$\text{Sum of deviation of observations from 4 is 72}$

$\text{Sum of deviation of observations from 7 is -3}$

$\textbf{To find:}$

$\text{Number of observations and mean}$

$\textbf{Solution:}$

$\text{Let n be the number of observations}$

$\text{As per given data,}$

$(x_1-4)+(x_2-4)+......+(x_n-4)=72$

$(x_1+x_2+.....+x_n)-4\,n=72$

$(x_1+x_2+.....+x_n)=72+4\,n$

$\Sigma\,x_i=72+4\,n$ ........(1)

$\text{Also,}$

$(x_1-7)+(x_2-7)+......+(x_n-7)=-3$

$(x_1+x_2+.....+x_n)-7\,n=-3$

$(x_1+x_2+.....+x_n)=7\,n-3$

$\Sigma\,x_i=7\,n-3$ ........(2)

$\text{From (1) and (2), we get}$

$72+4\,n=7\,n-3$

$72+3=7\,n-4\,n$

$75=3\,n$

$\n=\dfrac{75}{3}$

$\implies\boxed{\bf\,n=25}$

$\text{Now,}$

$\text{Mean}=\dfrac{\Sigma\,x_i}{n}$

$\text{Mean}=\dfrac{7\,n-3}{n}$

$\text{Mean}=\dfrac{7(25)-3}{25}$

$\text{Mean}=\dfrac{175-3}{25}$

$\text{Mean}=\dfrac{172}{25}$

$\implies\boxed{\textbf{Mean}=\bf\,6.88}$

$\textbf{Answer:}$

$\textbf{Number of observations is 25}$

$\textbf{and mean 6.88}$

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