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Home > Statistical Methods calculators > Population Skewness, Kurtosis for ungrouped data example
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Population Kurtosis Example for ungrouped data
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- Formula & Example
- Population Skewness Example
- Population Kurtosis Example
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Other related methods
- Mean, Median and Mode
- Quartile, Decile, Percentile, Octile, Quintile
- Population Variance, Standard deviation and coefficient of variation
- Sample Variance, Standard deviation and coefficient of variation
- Population Skewness, Kurtosis
- Sample Skewness, Kurtosis
- Geometric mean, Harmonic mean
- Mean deviation, Quartile deviation, Decile deviation, Percentile deviation
- Five number summary
- Box and Whisker Plots
- Construct an ungrouped frequency distribution table
- Construct a grouped frequency distribution table
- Maximum, Minimum
- Sum, Length
- Range, Mid Range
- Stem and leaf plot
- Ascending order, Descending order
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3. Population Kurtosis Example
1. Calculate Population 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 |
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(1216/10)`
`=sqrt(121.6)`
`=11.0272`
Population Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`
`=317284/(10*(11.0272)^4)`
`=317284/(10*14786.56)`
`=2.1458`
2. Calculate Population 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 |
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(3000/8)`
`=sqrt(375)`
`=19.3649`
Population Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`
`=3060000/(8*(19.3649)^4)`
`=3060000/(8*140625)`
`=2.72`
3. Calculate Population Kurtosis from the following data
`73,70,71,73,68,67,69,72,76,71`
Solution:
Kurtosis :
Mean `bar x=(sum x)/n`
`=(73+70+71+73+68+67+69+72+76+71)/10`
`=710/10`
`=71`
| `x` | `(x - bar x)`
`= (x - 71)` | `(x - bar x)^2`
`= (x - 71)^2` | `(x - bar x)^3`
`= (x - 71)^3` | `(x - bar x)^4`
`= (x - 71)^4` | | 73 | 2 | 4 | 8 | 16 | | 70 | -1 | 1 | -1 | 1 | | 71 | 0 | 0 | 0 | 0 | | 73 | 2 | 4 | 8 | 16 | | 68 | -3 | 9 | -27 | 81 | | 67 | -4 | 16 | -64 | 256 | | 69 | -2 | 4 | -8 | 16 | | 72 | 1 | 1 | 1 | 1 | | 76 | 5 | 25 | 125 | 625 | | 71 | 0 | 0 | 0 | 0 | | --- | --- | --- | --- | --- | | 710 | 0 | 64 | 42 | 1012 |
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(64/10)`
`=sqrt(6.4)`
`=2.5298`
Population Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`
`=1012/(10*(2.5298)^4)`
`=1012/(10*40.96)`
`=2.4707`
This material is intended as a summary. Use your textbook for detail explanation.
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