Question

Find the mean of the following data $30,32,24,34,26.28,30,35,33,25$
(i) Show that the sum of the deviations of all the given observations from the mean is zero.
(ii) Find the median of the given data.

Answer 1

Arranging given numbers in ascending order,

$24,25,26,28,30,30,32,33,34,35$

$Mean\left(\overline{x}\right)=\frac{24+25+26+28+30+30+32+33+34+35}{10}=\frac{297}{10}=29.7$

$\left(i\right)$

$\Rightarrow$ $(24-29.7)+(25-29.7)+(26-29.7)+(28-29.7)+(30-29.7)+(30-29.7)+(32-29.7)+(33-29.7)$$+(34-29.7)+(35-29.7)$

$\Rightarrow$ $-5.7-4.7-3.7-1.7+0.3+0.3+2.3+3.3+4.3+5.3$

$\Rightarrow$ $0$

Hence, we proved that the sum of the deviations of all the given observations from the mean is zero.

$\left(ii\right)$

Number of observation $=10$

$Median=\frac{1}{2}[{\left(\frac{10}{2}\right)}^{th}obs+{(\frac{10}{2}+1)}^{th}obs]$

$Median=\frac{1}{2}[5^{th}obs+6^{th}obs]$

$5^{th}=30$ and $6^{th}obs=30$

$\therefore$ $Median=\frac{30+30}{2}=30$