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 MomentsFirst 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 originFirst 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 MomentsFirst 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 originFirst 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 MomentsFirst 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 originFirst 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)
This material is intended as a summary. Use your textbook for detail explanation.
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