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

Let `barx` be the mean of n number of observation x₁, x₂, x₃, ...., x_n

Mean = `[ x_1 + x_2 + x_3 + ....... + x_n ]/n`
Therefore,
Mean of given data = `[ 30 + 32 + 24 + 34 + 26 + 28 + 30 + 35 + 33 + 25 ]/10`

= `297/10`
= 29.7

(i) Let us tabulate the observations and their deviations from the mean

Observation xi	Deviation `x_i - barx`
30	0.3
32	2.3
24	- 5.7
34	4.3
26	- 3.7
28	- 1.7
30	0.3
35	5.3
33	3.3
25	- 4.7
Total	0

From the table, it is clear that the sum of the deviations from the mean is Zero.

(ii) Consider the given data :
30, 32, 24, 34, 26, 28, 30, 35, 33, 25.
Let us rewrite the above data in ascending order
24, 25, 26, 28, 30, 30, 32, 33, 34, 35.
There are 10 observations, which is even.
Therefore,
Median = `1/2[ (n/2)^"th" "term" + ( n/2 + 1 )^"th" "term" ]`

= `1/2[ (10/2)^"th" "term" + ( 10/2 + 1)^"th" "term" ]`

= `1/2[ (5)^"th" "term" + ( 10/2 + 1)^"th" "term" ]`

= `1/2[ 5^"th" "term" + ( 5 + 1 )^"th" "term" ]`

= `1/2[ 5^"th" "term" + 6^"th" "term" ]`

= `1/2[ 30 + 30 ]` 

= `1/2[ 60 ]` 
= 30.