Population Skewness, Kurtosis for ungrouped data Formula & Example

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Population Skewness, Kurtosis for ungrouped data Formula & Example ( Enter your problem )

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4. Sample Variance, Standard deviation and coefficient of variation
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2. Population Skewness Example
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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`
---	---	---	---
49	0	122	120

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`
---	---	---	---
880	0	1216	2430

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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