1. Calculate Sample Kurtosis from the following data
`85,96,76,108,85,80,100,85,70,95`Solution:Kurtosis :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` | `(x - bar x)^4`
`=(x-88)^4` |
| 85 | -3 | 9 | -27 | 81 |
| 96 | 8 | 64 | 512 | 4096 |
| 76 | -12 | 144 | -1728 | 20736 |
| 108 | 20 | 400 | 8000 | 160000 |
| 85 | -3 | 9 | -27 | 81 |
| 80 | -8 | 64 | -512 | 4096 |
| 100 | 12 | 144 | 1728 | 20736 |
| 85 | -3 | 9 | -27 | 81 |
| 70 | -18 | 324 | -5832 | 104976 |
| 95 | 7 | 49 | 343 | 2401 |
| --- | --- | --- | --- | --- |
| `880` | `0` | `1216` | `2430` | `317284` |
Sample Standard deviation `S = sqrt((sum (x - bar x)^2)/(n-1))`
`=sqrt(1216/9)`
`=sqrt(135.1111)`
`=11.6237`
Sample Kurtosis `= (sum (x - bar x)^4)/((n-1)*S^4)`
`=317284/(9*(11.6237)^4)`
`=317284/(9*18255.0123)`
`=1.9312`
2. Calculate Sample Kurtosis from the following data
`10,50,30,20,10,20,70,30`Solution:Kurtosis :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` | `(x - bar x)^4`
`=(x-30)^4` |
| 10 | -20 | 400 | -8000 | 160000 |
| 50 | 20 | 400 | 8000 | 160000 |
| 30 | 0 | 0 | 0 | 0 |
| 20 | -10 | 100 | -1000 | 10000 |
| 10 | -20 | 400 | -8000 | 160000 |
| 20 | -10 | 100 | -1000 | 10000 |
| 70 | 40 | 1600 | 64000 | 2560000 |
| 30 | 0 | 0 | 0 | 0 |
| --- | --- | --- | --- | --- |
| `240` | `0` | `3000` | `54000` | `3060000` |
Sample Standard deviation `S = sqrt((sum (x - bar x)^2)/(n-1))`
`=sqrt(3000/7)`
`=sqrt(428.5714)`
`=20.702`
Sample Kurtosis `= (sum (x - bar x)^4)/((n-1)*S^4)`
`=3060000/(7*(20.702)^4)`
`=3060000/(7*183673.4694)`
`=2.38`
3. Calculate Sample Kurtosis from the following data
`10,50,30,20,10,20,70,30`Solution:Kurtosis :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` | `(x - bar x)^4`
`=(x-30)^4` |
| 10 | -20 | 400 | -8000 | 160000 |
| 50 | 20 | 400 | 8000 | 160000 |
| 30 | 0 | 0 | 0 | 0 |
| 20 | -10 | 100 | -1000 | 10000 |
| 10 | -20 | 400 | -8000 | 160000 |
| 20 | -10 | 100 | -1000 | 10000 |
| 70 | 40 | 1600 | 64000 | 2560000 |
| 30 | 0 | 0 | 0 | 0 |
| --- | --- | --- | --- | --- |
| `240` | `0` | `3000` | `54000` | `3060000` |
Sample Standard deviation `S = sqrt((sum (x - bar x)^2)/(n-1))`
`=sqrt(3000/7)`
`=sqrt(428.5714)`
`=20.702`
Sample Kurtosis `= (sum (x - bar x)^4)/((n-1)*S^4)`
`=3060000/(7*(20.702)^4)`
`=3060000/(7*183673.4694)`
`=2.38`
This material is intended as a summary. Use your textbook for detail explanation.
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