1. Calculate Moment about mean from the following grouped data
Solution:Moments :Mean `bar x=(sum f x)/n`
`=55/25`
`=2.2`
`x`
`(2)` | `f`
`(3)` | `f*x`
`(4)=(2)xx(3)` | `(x-bar x)`
`(5)` | `f*(x-bar x)`
`(6)=(3)xx(5)` | `f*(x-bar x)^2`
`(7)=(5)xx(6)` | `f*(x-bar x)^3`
`(8)=(5)xx(7)` | `f*(x-bar x)^4`
`(9)=(5)xx(8)` |
| 0 | 1 | 0 | -2.2 | -2.2 | 4.84 | -10.648 | 23.4256 |
| 1 | 5 | 5 | -1.2 | -6 | 7.2 | -8.64 | 10.368 |
| 2 | 10 | 20 | -0.2 | -2 | 0.4 | -0.08 | 0.016 |
| 3 | 6 | 18 | 0.8 | 4.8 | 3.84 | 3.072 | 2.4576 |
| 4 | 3 | 12 | 1.8 | 5.4 | 9.72 | 17.496 | 31.4928 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| -- | `n=25` | `sum f*x=55` | -- | `=0` | `=26` | `=1.2` | `=67.76` |
Now, calculate Central MomentsFirst Central Moment`m_1=(sum f*(x-bar x))/n`
`=(0)/(25)`
`=0`
Second Central Moment`m_2=(sum f*(x-bar x)^2)/n`
`=(26)/(25)`
`=1.04`
Third Central Moment`m_3=(sum f*(x-bar x)^3)/n`
`=(1.2)/(25)`
`=0.048`
Fourth Central Moment`m_4=(sum f*(x-bar x)^4)/n`
`=(67.76)/(25)`
`=2.7104`
Skewness `beta_1=(m_3)^2/(m_2)^3`
`=(0.048)^2/(1.04)^3`
`=(0.0023)/(1.1249)`
`=0.002`
Kurtosis `beta_2=(m_4)/(m_2)^2`
`=(2.7104)/(1.04)^2`
`=(2.7104)/(1.0816)`
`=2.5059`
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 mean from the following grouped data
| X | Frequency |
| 10 | 3 |
| 11 | 12 |
| 12 | 18 |
| 13 | 12 |
| 14 | 3 |
Solution:Moments :Mean `bar x=(sum f x)/n`
`=576/48`
`=12`
`x`
`(2)` | `f`
`(3)` | `f*x`
`(4)=(2)xx(3)` | `(x-bar x)`
`(5)` | `f*(x-bar x)`
`(6)=(3)xx(5)` | `f*(x-bar x)^2`
`(7)=(5)xx(6)` | `f*(x-bar x)^3`
`(8)=(5)xx(7)` | `f*(x-bar x)^4`
`(9)=(5)xx(8)` |
| 10 | 3 | 30 | -2 | -6 | 12 | -24 | 48 |
| 11 | 12 | 132 | -1 | -12 | 12 | -12 | 12 |
| 12 | 18 | 216 | 0 | 0 | 0 | 0 | 0 |
| 13 | 12 | 156 | 1 | 12 | 12 | 12 | 12 |
| 14 | 3 | 42 | 2 | 6 | 12 | 24 | 48 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| -- | `n=48` | `sum f*x=576` | -- | `=0` | `=48` | `=0` | `=120` |
Now, calculate Central MomentsFirst Central Moment`m_1=(sum f*(x-bar x))/n`
`=(0)/(48)`
`=0`
Second Central Moment`m_2=(sum f*(x-bar x)^2)/n`
`=(48)/(48)`
`=1`
Third Central Moment`m_3=(sum f*(x-bar x)^3)/n`
`=(0)/(48)`
`=0`
Fourth Central Moment`m_4=(sum f*(x-bar x)^4)/n`
`=(120)/(48)`
`=2.5`
Skewness `beta_1=(m_3)^2/(m_2)^3`
`=(0)^2/(1)^3`
`=(0)/(1)`
`=0`
Kurtosis `beta_2=(m_4)/(m_2)^2`
`=(2.5)/(1)^2`
`=(2.5)/(1)`
`=2.5`
Moment coefficient of skewness`beta_1=0` : The distribution is perfectly symmetrical (like a normal distribution).
Moment coefficient of kurtosis`beta_2<3` : platykurtic (flatter with lighter tails)
3. Calculate Moment about mean from the following grouped data
| Class | Frequency |
| 2 - 4 | 3 |
| 4 - 6 | 4 |
| 6 - 8 | 2 |
| 8 - 10 | 1 |
Solution:Moments :Mean `bar x=(sum f x)/(sum f)`
`=52/10`
`=5.2`
Class
`(1)` | Mid value (`x`)
`(2)` | `f`
`(3)` | `f*x`
`(4)=(2)xx(3)` | `(x-bar x)`
`(5)` | `f*(x-bar x)`
`(6)=(3)xx(5)` | `f*(x-bar x)^2`
`(7)=(5)xx(6)` | `f*(x-bar x)^3`
`(8)=(5)xx(7)` | `f*(x-bar x)^4`
`(9)=(5)xx(8)` |
| 2 - 4 | 3 | 3 | 9 | -2.2 | -6.6 | 14.52 | -31.944 | 70.2768 |
| 4 - 6 | 5 | 4 | 20 | -0.2 | -0.8 | 0.16 | -0.032 | 0.0064 |
| 6 - 8 | 7 | 2 | 14 | 1.8 | 3.6 | 6.48 | 11.664 | 20.9952 |
| 8 - 10 | 9 | 1 | 9 | 3.8 | 3.8 | 14.44 | 54.872 | 208.5136 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| -- | -- | `n=10` | `sum f*x=52` | -- | `=0` | `=35.6` | `=34.56` | `=299.792` |
Now, calculate Central MomentsFirst Central Moment`m_1=(sum f*(x-bar x))/n`
`=(0)/(10)`
`=0`
Second Central Moment`m_2=(sum f*(x-bar x)^2)/n`
`=(35.6)/(10)`
`=3.56`
Third Central Moment`m_3=(sum f*(x-bar x)^3)/n`
`=(34.56)/(10)`
`=3.456`
Fourth Central Moment`m_4=(sum f*(x-bar x)^4)/n`
`=(299.792)/(10)`
`=29.9792`
Skewness `beta_1=(m_3)^2/(m_2)^3`
`=(3.456)^2/(3.56)^3`
`=(11.9439)/(45.118)`
`=0.2647`
Kurtosis `beta_2=(m_4)/(m_2)^2`
`=(29.9792)/(3.56)^2`
`=(29.9792)/(12.6736)`
`=2.3655`
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)
4. Calculate Moment about mean from the following grouped data
| Class | Frequency |
| 0 - 2 | 5 |
| 2 - 4 | 16 |
| 4 - 6 | 13 |
| 6 - 8 | 7 |
| 8 - 10 | 5 |
| 10 - 12 | 4 |
Solution:Moments :Mean `bar x=(sum f x)/(sum f)`
`=256/50`
`=5.12`
Class
`(1)` | Mid value (`x`)
`(2)` | `f`
`(3)` | `f*x`
`(4)=(2)xx(3)` | `(x-bar x)`
`(5)` | `f*(x-bar x)`
`(6)=(3)xx(5)` | `f*(x-bar x)^2`
`(7)=(5)xx(6)` | `f*(x-bar x)^3`
`(8)=(5)xx(7)` | `f*(x-bar x)^4`
`(9)=(5)xx(8)` |
| 0 - 2 | 1 | 5 | 5 | -4.12 | -20.6 | 84.872 | -349.6726 | 1440.6513 |
| 2 - 4 | 3 | 16 | 48 | -2.12 | -33.92 | 71.9104 | -152.45 | 323.1941 |
| 4 - 6 | 5 | 13 | 65 | -0.12 | -1.56 | 0.1872 | -0.0225 | 0.0027 |
| 6 - 8 | 7 | 7 | 49 | 1.88 | 13.16 | 24.7408 | 46.5127 | 87.4439 |
| 8 - 10 | 9 | 5 | 45 | 3.88 | 19.4 | 75.272 | 292.0554 | 1133.1748 |
| 10 - 12 | 11 | 4 | 44 | 5.88 | 23.52 | 138.2976 | 813.1899 | 4781.5565 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| -- | -- | `n=50` | `sum f*x=256` | -- | `=0` | `=395.28` | `=649.6128` | `=7766.0233` |
Now, calculate Central MomentsFirst Central Moment`m_1=(sum f*(x-bar x))/n`
`=(0)/(50)`
`=0`
Second Central Moment`m_2=(sum f*(x-bar x)^2)/n`
`=(395.28)/(50)`
`=7.9056`
Third Central Moment`m_3=(sum f*(x-bar x)^3)/n`
`=(649.6128)/(50)`
`=12.9923`
Fourth Central Moment`m_4=(sum f*(x-bar x)^4)/n`
`=(7766.0233)/(50)`
`=155.3205`
Skewness `beta_1=(m_3)^2/(m_2)^3`
`=(12.9923)^2/(7.9056)^3`
`=(168.7987)/(494.0882)`
`=0.3416`
Kurtosis `beta_2=(m_4)/(m_2)^2`
`=(155.3205)/(7.9056)^2`
`=(155.3205)/(62.4985)`
`=2.4852`
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.
Any bug, improvement, feedback then
