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

Solution:
Moments :
`A=20`

`x`	`(x-A)` `=(x-20)`	`(x-A)^2` `=(x-20)^2`	`(x-A)^3` `=(x-20)^3`	`(x-A)^4` `=(x-20)^4`
10	-10	100	-1000	10000
50	30	900	27000	810000
30	10	100	1000	10000
20	0	0	0	0
10	-10	100	-1000	10000
20	0	0	0	0
70	50	2500	125000	6250000
30	10	100	1000	10000
---	---	---	---	---
`240`	`80`	`3800`	`152000`	`7100000`

Now, calculate Raw Moments

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

`=(80)/(8)`

`=10`

---

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

`=(3800)/(8)`

`=475`

---

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

`=(152000)/(8)`

`=19000`

---

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

`=(7100000)/(8)`

`=887500`

---

Find Central moments using Moments about the value 20

First Central Moment
`m_1=0`

---

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

`=475-10^2`

`=475-100`

`=375`

---

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

`=19000-3*475*10+2*10^3`

`=19000-14250+2000`

`=6750`

---

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

`=887500-4*19000*10+6*475*10^2-3*10^4`

`=887500-760000+285000-30000`

`=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 the value 90 from the following data
`85,96,76,108,85,80,100,85,70,95`

Solution:
Moments :
`A=90`

`x`	`(x-A)` `=(x-90)`	`(x-A)^2` `=(x-90)^2`	`(x-A)^3` `=(x-90)^3`	`(x-A)^4` `=(x-90)^4`
85	-5	25	-125	625
96	6	36	216	1296
76	-14	196	-2744	38416
108	18	324	5832	104976
85	-5	25	-125	625
80	-10	100	-1000	10000
100	10	100	1000	10000
85	-5	25	-125	625
70	-20	400	-8000	160000
95	5	25	125	625
---	---	---	---	---
`880`	`-20`	`1256`	`-4946`	`327188`

Now, calculate Raw Moments

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

`=(-20)/(10)`

`=-2`

---

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

`=(1256)/(10)`

`=125.6`

---

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

`=(-4946)/(10)`

`=-494.6`

---

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

`=(327188)/(10)`

`=32718.8`

---

Find Central moments using Moments about the value 90

First Central Moment
`m_1=0`

---

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

`=125.6-(-2)^2`

`=125.6-4`

`=121.6`

---

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

`=(-494.6)-3*125.6*(-2)+2*(-2)^3`

`=(-494.6)+753.6-16`

`=243`

---

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

`=32718.8-4*(-494.6)*(-2)+6*125.6*(-2)^2-3*(-2)^4`

`=32718.8-3956.8+3014.4-48`

`=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 the value 10 from the following data
`3,23,13,11,15,5,4,2`

Solution:
Moments :
`A=10`

`x`	`(x-A)` `=(x-10)`	`(x-A)^2` `=(x-10)^2`	`(x-A)^3` `=(x-10)^3`	`(x-A)^4` `=(x-10)^4`
3	-7	49	-343	2401
23	13	169	2197	28561
13	3	9	27	81
11	1	1	1	1
15	5	25	125	625
5	-5	25	-125	625
4	-6	36	-216	1296
2	-8	64	-512	4096
---	---	---	---	---
`76`	`-4`	`378`	`1154`	`37686`

Now, calculate Raw Moments

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

`=(-4)/(8)`

`=-0.5`

---

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

`=(378)/(8)`

`=47.25`

---

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

`=(1154)/(8)`

`=144.25`

---

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

`=(37686)/(8)`

`=4710.75`

---

Find Central moments using Moments about the value 10

First Central Moment
`m_1=0`

---

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

`=47.25-(-0.5)^2`

`=47.25-0.25`

`=47`

---

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

`=144.25-3*47.25*(-0.5)+2*(-0.5)^3`

`=144.25+70.875-0.25`

`=214.875`

---

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

`=4710.75-4*144.25*(-0.5)+6*47.25*(-0.5)^2-3*(-0.5)^4`

`=4710.75+288.5+70.875-0.1875`

`=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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