Question

Find the mean of the following data:
30, 32, 24, 34, 26, 28, 30, 35, 33, 25.
Show that the sum of the deviations of all the given observations from the mean is zero.

Answer 1

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

deviation $d_i=\overline{x}-x_i$

$d_1=29.7-30=-0.3$

$d_2=29.7-32=-2.3$

$d_3=29.7-24=5.7$

$d_4=29.7-34=-5.7$

$d_4=29.7-26=-4.3$

$d_5=29.7-26=3.7$

$d_6=29.7-28=1.7$

$d_7=29.7-30=-0.3$

$d_8=29.7-35=-5.3$

$d_9=29.7-33=-3.3$

$d_{10}=29.7-25=4.7$

Sum of deviations $=\sum d_i$

$=\left(0.3\right)+(-2.3)+5.7+(-4.3)+3.7+1.7+(-0.3)+(-5.3)+(-3.3)+4.7$

Sum of deviations from mean $=6$

Hence proved