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 MomentsFirst 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 20First 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 MomentsFirst 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 90First 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 MomentsFirst 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 10First 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)
This material is intended as a summary. Use your textbook for detail explanation.
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