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Home > Statistical Methods calculators > Population Skewness, Kurtosis for ungrouped data example
Population Skewness, Kurtosis for ungrouped data Formula & Example ( Enter your problem )
( Enter your problem )
  1. Formula & Example
  2. Population Skewness Example
  3. Population Kurtosis Example
 		Other related methods
  1. Mean, Median and Mode
  2. Quartile, Decile, Percentile, Octile, Quintile
  3. Population Variance, Standard deviation and coefficient of variation
  4. Sample Variance, Standard deviation and coefficient of variation
  5. Population Skewness, Kurtosis
  6. Sample Skewness, Kurtosis
  7. Geometric mean, Harmonic mean
  8. Mean deviation, Quartile deviation, Decile deviation, Percentile deviation
  9. Five number summary
  10. Box and Whisker Plots
  11. Construct an ungrouped frequency distribution table
  12. Construct a grouped frequency distribution table
  13. Maximum, Minimum
  14. Sum, Length
  15. Range, Mid Range
  16. Stem and leaf plot
  17. Ascending order, Descending order
4. Sample Variance, Standard deviation and coefficient of variation
(Previous method)		2. Population Skewness Example
(Next example)

1. Formula & Example


Formula
1. Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
2. Skewness `= (sum(x - bar x)^3)/(n*S^3)`
3. Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`
Examples
1. Calculate Population Skewness, Population Kurtosis from the following data
3,13,11,11,5,4,2

Solution:
Mean `bar x = (sum x)/n`

`=(3 + 13 + 11 + 11 + 5 + 4 + 2)/7`

`=49/7`

`=7`

`x``(x - bar x)`
`= (x - 7)`		`(x - bar x)^2`
`= (x - 7)^2`		`(x - bar x)^3`
`= (x - 7)^3`
3 -4 `(3-7)=-4`
`(x - 7)`		 16 `(3-7)^2=16`
`(x - 7)^2`		 -64 `(3-7)^3=-64`
`(x - 7)^3`
13 6 `(13-7)=6`
`(x - 7)`		 36 `(13-7)^2=36`
`(x - 7)^2`		 216 `(13-7)^3=216`
`(x - 7)^3`
11 4 `(11-7)=4`
`(x - 7)`		 16 `(11-7)^2=16`
`(x - 7)^2`		 64 `(11-7)^3=64`
`(x - 7)^3`
11 4 `(11-7)=4`
`(x - 7)`		 16 `(11-7)^2=16`
`(x - 7)^2`		 64 `(11-7)^3=64`
`(x - 7)^3`
5 -2 `(5-7)=-2`
`(x - 7)`		 4 `(5-7)^2=4`
`(x - 7)^2`		 -8 `(5-7)^3=-8`
`(x - 7)^3`
4 -3 `(4-7)=-3`
`(x - 7)`		 9 `(4-7)^2=9`
`(x - 7)^2`		 -27 `(4-7)^3=-27`
`(x - 7)^3`
2 -5 `(2-7)=-5`
`(x - 7)`		 25 `(2-7)^2=25`
`(x - 7)^2`		 -125 `(2-7)^3=-125`
`(x - 7)^3`
------------
490122120


Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`

`=sqrt(122/7)`

`=sqrt(17.4286)`

`=4.1748`


Skewness `= (sum(x - bar x)^3)/(n*S^3)`

`=120/(7*(4.1748)^3)`

`=120/(7*72.76)`

`=0.2356`


Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`

`=2786/(7*(4.1748)^4)`

`=2786/(7*303.7551)`

`=1.3103`
2. Calculate Population Skewness, Population Kurtosis from the following data
85,96,76,108,85,80,100,85,70,95

Solution:
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`
85 -3 `(85-88)=-3`
`(x - 88)`		 9 `(85-88)^2=9`
`(x - 88)^2`		 -27 `(85-88)^3=-27`
`(x - 88)^3`
96 8 `(96-88)=8`
`(x - 88)`		 64 `(96-88)^2=64`
`(x - 88)^2`		 512 `(96-88)^3=512`
`(x - 88)^3`
76 -12 `(76-88)=-12`
`(x - 88)`		 144 `(76-88)^2=144`
`(x - 88)^2`		 -1728 `(76-88)^3=-1728`
`(x - 88)^3`
108 20 `(108-88)=20`
`(x - 88)`		 400 `(108-88)^2=400`
`(x - 88)^2`		 8000 `(108-88)^3=8000`
`(x - 88)^3`
85 -3 `(85-88)=-3`
`(x - 88)`		 9 `(85-88)^2=9`
`(x - 88)^2`		 -27 `(85-88)^3=-27`
`(x - 88)^3`
80 -8 `(80-88)=-8`
`(x - 88)`		 64 `(80-88)^2=64`
`(x - 88)^2`		 -512 `(80-88)^3=-512`
`(x - 88)^3`
100 12 `(100-88)=12`
`(x - 88)`		 144 `(100-88)^2=144`
`(x - 88)^2`		 1728 `(100-88)^3=1728`
`(x - 88)^3`
85 -3 `(85-88)=-3`
`(x - 88)`		 9 `(85-88)^2=9`
`(x - 88)^2`		 -27 `(85-88)^3=-27`
`(x - 88)^3`
70 -18 `(70-88)=-18`
`(x - 88)`		 324 `(70-88)^2=324`
`(x - 88)^2`		 -5832 `(70-88)^3=-5832`
`(x - 88)^3`
95 7 `(95-88)=7`
`(x - 88)`		 49 `(95-88)^2=49`
`(x - 88)^2`		 343 `(95-88)^3=343`
`(x - 88)^3`
------------
880012162430


Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`

`=sqrt(1216/10)`

`=sqrt(121.6)`

`=11.0272`


Skewness `= (sum(x - bar x)^3)/(n*S^3)`

`=2430/(10*(11.0272)^3)`

`=2430/(10*1340.9123)`

`=0.1812`


Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`

`=317284/(10*(11.0272)^4)`

`=317284/(10*14786.56)`

`=2.1458`

This material is intended as a summary. Use your textbook for detail explanation.
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4. Sample Variance, Standard deviation and coefficient of variation
(Previous method)		2. Population Skewness Example
(Next example)


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