1. Calculate Percentile deviation from the following data
`10,50,30,20,10,20,70,30`

Solution:
Percentile deviation :
Arranging Observations in the ascending order, We get :
`10,10,20,20,30,30,50,70`

Here, `n = 8`

`P_10 = ((10(n+1))/100)^(th)` value of the observation

`=((10*9)/100)^(th)` value of the observation

`=(0.9)^(th)` value of the observation

`=0^(th)` observation ` + 0.9 [1^(st) - 0^(th)]`

`=0 + 0.9 [10 - 0]`

`=0 + 0.9 (10)`

`=0 + 9`

`=9`

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`P_90 = ((90(n+1))/100)^(th)` value of the observation

`=((90*9)/100)^(th)` value of the observation

`=(8.1)^(th)` value of the observation

`=8^(th)` observation ` + 0.1 [9^(th) - 8^(th)]`

`=70 + 0.1 [0 - 70]`

`=70 + 0.1 (-70)`

`=70 + -7`

`=63`

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Inter Percentile range `=P_90 - P_10=63-9=54`

Percentile deviation `=(P_90 - P_10)/2=(63-9)/2=54/2=27`

Coefficient of Percentile deviation `=(P_90 - P_10)/(P_90 + P_10)=(63-9)/(63+9)=54/72=0.75`

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2. Calculate Percentile deviation from the following data
`85,96,76,108,85,80,100,85,70,95`

Solution:
Percentile deviation :
Arranging Observations in the ascending order, We get :
`70,76,80,85,85,85,95,96,100,108`

Here, `n = 10`

`P_10 = ((10(n+1))/100)^(th)` value of the observation

`=((10*11)/100)^(th)` value of the observation

`=(1.1)^(th)` value of the observation

`=1^(st)` observation ` + 0.1 [2^(nd) - 1^(st)]`

`=70 + 0.1 [76 - 70]`

`=70 + 0.1 (6)`

`=70 + 0.6`

`=70.6`

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`P_90 = ((90(n+1))/100)^(th)` value of the observation

`=((90*11)/100)^(th)` value of the observation

`=(9.9)^(th)` value of the observation

`=9^(th)` observation ` + 0.9 [10^(th) - 9^(th)]`

`=100 + 0.9 [108 - 100]`

`=100 + 0.9 (8)`

`=100 + 7.2`

`=107.2`

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Inter Percentile range `=P_90 - P_10=107.2-70.6=36.6`

Percentile deviation `=(P_90 - P_10)/2=(107.2-70.6)/2=36.6/2=18.3`

Coefficient of Percentile deviation `=(P_90 - P_10)/(P_90 + P_10)=(107.2-70.6)/(107.2+70.6)=36.6/177.8=0.2058`

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3. Calculate Percentile deviation from the following data
`73,70,71,73,68,67,69,72,76,71`

Solution:
Percentile deviation :
Arranging Observations in the ascending order, We get :
`67,68,69,70,71,71,72,73,73,76`

Here, `n = 10`

`P_10 = ((10(n+1))/100)^(th)` value of the observation

`=((10*11)/100)^(th)` value of the observation

`=(1.1)^(th)` value of the observation

`=1^(st)` observation ` + 0.1 [2^(nd) - 1^(st)]`

`=67 + 0.1 [68 - 67]`

`=67 + 0.1 (1)`

`=67 + 0.1`

`=67.1`

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`P_90 = ((90(n+1))/100)^(th)` value of the observation

`=((90*11)/100)^(th)` value of the observation

`=(9.9)^(th)` value of the observation

`=9^(th)` observation ` + 0.9 [10^(th) - 9^(th)]`

`=73 + 0.9 [76 - 73]`

`=73 + 0.9 (3)`

`=73 + 2.7`

`=75.7`

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Inter Percentile range `=P_90 - P_10=75.7-67.1=8.6`

Percentile deviation `=(P_90 - P_10)/2=(75.7-67.1)/2=8.6/2=4.3`

Coefficient of Percentile deviation `=(P_90 - P_10)/(P_90 + P_10)=(75.7-67.1)/(75.7+67.1)=8.6/142.8=0.0602`

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