1. Calculate Population Skewness from the following data
`85,96,76,108,85,80,100,85,70,95`

Solution:
Skewness :
Mean `bar x=(sum x)/n`

`=(85+96+76+108+85+80+100+85+70+95)/10`

`=880/10`

`=88`

`x`	`(x - bar x)` `=(x-88)`	`(x - bar x)^2` `=(x-88)^2`	`(x - bar x)^3` `=(x-88)^3`
85	-3	9	-27
96	8	64	512
76	-12	144	-1728
108	20	400	8000
85	-3	9	-27
80	-8	64	-512
100	12	144	1728
85	-3	9	-27
70	-18	324	-5832
95	7	49	343
---	---	---	---
`880`	`0`	`1216`	`2430`

Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`

`=sqrt(1216/10)`

`=sqrt(121.6)`

`=11.0272`

---

Population Skewness `= (sum (x - bar x)^3)/(n*S^3)`

`=2430/(10*(11.0272)^3)`

`=2430/(10*1340.9123)`

`=0.1812`

---

2. Calculate Population Skewness from the following data
`10,50,30,20,10,20,70,30`

Solution:
Skewness :
Mean `bar x=(sum x)/n`

`=(10+50+30+20+10+20+70+30)/8`

`=240/8`

`=30`

`x`	`(x - bar x)` `=(x-30)`	`(x - bar x)^2` `=(x-30)^2`	`(x - bar x)^3` `=(x-30)^3`
10	-20	400	-8000
50	20	400	8000
30	0	0	0
20	-10	100	-1000
10	-20	400	-8000
20	-10	100	-1000
70	40	1600	64000
30	0	0	0
---	---	---	---
`240`	`0`	`3000`	`54000`

Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`

`=sqrt(3000/8)`

`=sqrt(375)`

`=19.3649`

---

Population Skewness `= (sum (x - bar x)^3)/(n*S^3)`

`=54000/(8*(19.3649)^3)`

`=54000/(8*7261.8438)`

`=0.9295`

---

3. Calculate Population Skewness from the following data
`73,70,71,73,68,67,69,72,76,71`

Solution:
Skewness :
Mean `bar x=(sum x)/n`

`=(73+70+71+73+68+67+69+72+76+71)/10`

`=710/10`

`=71`

`x`	`(x - bar x)` `=(x-71)`	`(x - bar x)^2` `=(x-71)^2`	`(x - bar x)^3` `=(x-71)^3`
73	2	4	8
70	-1	1	-1
71	0	0	0
73	2	4	8
68	-3	9	-27
67	-4	16	-64
69	-2	4	-8
72	1	1	1
76	5	25	125
71	0	0	0
---	---	---	---
`710`	`0`	`64`	`42`

Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`

`=sqrt(64/10)`

`=sqrt(6.4)`

`=2.5298`

---

Population Skewness `= (sum (x - bar x)^3)/(n*S^3)`

`=42/(10*(2.5298)^3)`

`=42/(10*16.1909)`

`=0.2594`

---

---

---