1. Calculate Moment about origin from the following data
`10,50,30,20,10,20,70,30`

Solution:
Moments :

`x`	`x^2`	`x^3`	`x^4`
10	100	1000	10000
50	2500	125000	6250000
30	900	27000	810000
20	400	8000	160000
10	100	1000	10000
20	400	8000	160000
70	4900	343000	24010000
30	900	27000	810000
---	---	---	---
`240`	`10200`	`540000`	`32220000`

Now, calculate Raw Moments

First Raw Moment
`M_1=(sum x)/n`

`=(240)/(8)`

`=30`

---

Second Raw Moment
`M_2=(sum x^2)/n`

`=(10200)/(8)`

`=1275`

---

Third Raw Moment
`M_3=(sum x^3)/n`

`=(540000)/(8)`

`=67500`

---

Fourth Raw Moment
`M_4=(sum x^4)/n`

`=(32220000)/(8)`

`=4027500`

---

Find Central moments using Moments about origin

First Central Moment
`m_1=0`

---

Second Central Moment
`m_2=M_2-M_1^2`

`=1275-30^2`

`=1275-900`

`=375`

---

Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=67500-3*1275*30+2*30^3`

`=67500-114750+54000`

`=6750`

---

Fourth Central Moment
`m_4=M_4-4M_3M_1+6M_2M_1^2-3M_1^4`

`=4027500-4*67500*30+6*1275*30^2-3*30^4`

`=4027500-8100000+6885000-2430000`

`=382500`

---

Skewness `beta_1=(m_3)^2/(m_2)^3`

`=(6750)^2/(375)^3`

`=(45562500)/(52734375)`

`=0.864`

---

Kurtosis `beta_2=(m_4)/(m_2)^2`

`=(382500)/(375)^2`

`=(382500)/(140625)`

`=2.72`

---

Moment coefficient of skewness
`beta_1>0` : The distribution is positively skewed (a longer tail to the right).

Moment coefficient of kurtosis
`beta_2<3` : platykurtic (flatter with lighter tails)

---

2. Calculate Moment about origin from the following data
`85,96,76,108,85,80,100,85,70,95`

Solution:
Moments :

`x`	`x^2`	`x^3`	`x^4`
85	7225	614125	52200625
96	9216	884736	84934656
76	5776	438976	33362176
108	11664	1259712	136048896
85	7225	614125	52200625
80	6400	512000	40960000
100	10000	1000000	100000000
85	7225	614125	52200625
70	4900	343000	24010000
95	9025	857375	81450625
---	---	---	---
`880`	`78656`	`7138174`	`657368228`

Now, calculate Raw Moments

First Raw Moment
`M_1=(sum x)/n`

`=(880)/(10)`

`=88`

---

Second Raw Moment
`M_2=(sum x^2)/n`

`=(78656)/(10)`

`=7865.6`

---

Third Raw Moment
`M_3=(sum x^3)/n`

`=(7138174)/(10)`

`=713817.4`

---

Fourth Raw Moment
`M_4=(sum x^4)/n`

`=(657368228)/(10)`

`=65736822.8`

---

Find Central moments using Moments about origin

First Central Moment
`m_1=0`

---

Second Central Moment
`m_2=M_2-M_1^2`

`=7865.6-88^2`

`=7865.6-7744`

`=121.6`

---

Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=713817.4-3*7865.6*88+2*88^3`

`=713817.4-2076518.4+1362944`

`=243`

---

Fourth Central Moment
`m_4=M_4-4M_3M_1+6M_2M_1^2-3M_1^4`

`=65736822.8-4*713817.4*88+6*7865.6*88^2-3*88^4`

`=65736822.8-251263724.8+365467238.4-179908608`

`=31728.4`

---

Skewness `beta_1=(m_3)^2/(m_2)^3`

`=(243)^2/(121.6)^3`

`=(59049)/(1798045.696)`

`=0.0328`

---

Kurtosis `beta_2=(m_4)/(m_2)^2`

`=(31728.4)/(121.6)^2`

`=(31728.4)/(14786.56)`

`=2.1458`

---

Moment coefficient of skewness
`beta_1>0` : The distribution is positively skewed (a longer tail to the right).

Moment coefficient of kurtosis
`beta_2<3` : platykurtic (flatter with lighter tails)

---

3. Calculate Moment about origin from the following data
`3,23,13,11,15,5,4,2`

Solution:
Moments :

`x`	`x^2`	`x^3`	`x^4`
3	9	27	81
23	529	12167	279841
13	169	2197	28561
11	121	1331	14641
15	225	3375	50625
5	25	125	625
4	16	64	256
2	4	8	16
---	---	---	---
`76`	`1098`	`19294`	`374646`

Now, calculate Raw Moments

First Raw Moment
`M_1=(sum x)/n`

`=(76)/(8)`

`=9.5`

---

Second Raw Moment
`M_2=(sum x^2)/n`

`=(1098)/(8)`

`=137.25`

---

Third Raw Moment
`M_3=(sum x^3)/n`

`=(19294)/(8)`

`=2411.75`

---

Fourth Raw Moment
`M_4=(sum x^4)/n`

`=(374646)/(8)`

`=46830.75`

---

Find Central moments using Moments about origin

First Central Moment
`m_1=0`

---

Second Central Moment
`m_2=M_2-M_1^2`

`=137.25-9.5^2`

`=137.25-90.25`

`=47`

---

Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=2411.75-3*137.25*9.5+2*9.5^3`

`=2411.75-3911.625+1714.75`

`=214.875`

---

Fourth Central Moment
`m_4=M_4-4M_3M_1+6M_2M_1^2-3M_1^4`

`=46830.75-4*2411.75*9.5+6*137.25*9.5^2-3*9.5^4`

`=46830.75-91646.5+74320.875-24435.1875`

`=5069.9375`

---

Skewness `beta_1=(m_3)^2/(m_2)^3`

`=(214.875)^2/(47)^3`

`=(46171.2656)/(103823)`

`=0.4447`

---

Kurtosis `beta_2=(m_4)/(m_2)^2`

`=(5069.9375)/(47)^2`

`=(5069.9375)/(2209)`

`=2.2951`

---

Moment coefficient of skewness
`beta_1>0` : The distribution is positively skewed (a longer tail to the right).

Moment coefficient of kurtosis
`beta_2<3` : platykurtic (flatter with lighter tails)

---

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