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Find Population kurtosis {{10-20,15},{20-30,25},{30-40,20},{40-50,12},{50-60,8}} [ Calculator, Method and examples ]

Solution:
Your problem `->` Population kurtosis {{10-20,15},{20-30,25},{30-40,20},{40-50,12},{50-60,8}}


Kurtosis :
Mean `bar x=(sum f x)/(sum f)`

`=2530/80`

`=31.625`

Class
`(1)`
Mid value (`x`)
`(2)`
`f`
`(3)`
`f*x`
`(4)=(2)xx(3)`
`(x-bar x)`
`(5)`
`f*(x-bar x)^2`
`(6)=(3)xx(5)`
`f*(x-bar x)^3`
`(7)=(5)xx(6)`
10 - 20 15 `15=(10+20)/2`15 225 `225=15xx15`
`(4)=(2)xx(3)`
 -16.625 `|15-31.625|=-16.625`
`|x - 31.625|`
 4145.8594 `4145.8594=15xx-16.625xx-16.625`
`(6)=(2)xx(5)`
 -68924.9121 `-68924.9121=4145.8594xx-16.625`
`(7)=(2)xx(6)`
20 - 30 25 `25=(20+30)/2`25 625 `625=25xx25`
`(4)=(2)xx(3)`
 -6.625 `|25-31.625|=-6.625`
`|x - 31.625|`
 1097.2656 `1097.2656=25xx-6.625xx-6.625`
`(6)=(2)xx(5)`
 -7269.3848 `-7269.3848=1097.2656xx-6.625`
`(7)=(2)xx(6)`
30 - 40 35 `35=(30+40)/2`20 700 `700=20xx35`
`(4)=(2)xx(3)`
 3.375 `|35-31.625|=3.375`
`|x - 31.625|`
 227.8125 `227.8125=20xx3.375xx3.375`
`(6)=(2)xx(5)`
 768.8672 `768.8672=227.8125xx3.375`
`(7)=(2)xx(6)`
40 - 50 45 `45=(40+50)/2`12 540 `540=12xx45`
`(4)=(2)xx(3)`
 13.375 `|45-31.625|=13.375`
`|x - 31.625|`
 2146.6875 `2146.6875=12xx13.375xx13.375`
`(6)=(2)xx(5)`
 28711.9453 `28711.9453=2146.6875xx13.375`
`(7)=(2)xx(6)`
50 - 60 55 `55=(50+60)/2`8 440 `440=8xx55`
`(4)=(2)xx(3)`
 23.375 `|55-31.625|=23.375`
`|x - 31.625|`
 4371.125 `4371.125=8xx23.375xx23.375`
`(6)=(2)xx(5)`
 102175.0469 `102175.0469=4371.125xx23.375`
`(7)=(2)xx(6)`
---------------------
----`n=80``sum f*x=2530``--`=11988.75``=55461.5625`


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

`=sqrt(11988.75/80)`

`=sqrt(149.8594)`

`=12.2417`



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

`=3968995.2539/(80*(12.2417)^4)`

`=3968995.2539/(80*22457.8323)`

`=2.2091`








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