Question

Calculate the values of `Q_(1),Q_(3),D_(8)and P_(85)` - from the following data :
`{:("Wages (Rs.)",0-10,10-20,20-30,30-40,40-50),("Number of Workers"," "22," "38," "46," "35," "19):}`

Answer 1

`Q_(1)` = Size of `((N)/(4))`th item = Size of `((160)/(4))`th item
= Size of 40th item.
`Q_(1)` lies in 60th cumulative frequency . The class interval corresponding to cumulative frequency is 10-20.
Thus, `Q_(1)=l_(1)+(N/(2)-c.f.)/(f)xxi`
`=10+(160/(4)-22)/(38)xx10`
`=10+(40-22)/(38)xx10=10+(18)/(38)xx10`
`10+4.74=14.74` `Q_(3)` = Size of `3((N)/(4))` th item = Size of `3((160)/(4))` th item
= Size of 120th item
`Q_(3)` lies in 141 th cumulative frequency . the class interval corresponding to this cumulative frequency is 30-40.
Thus, `Q_(3)=l_(1)+3((N/(4))-c.f.)/(f)xxi`
`=30+3((160/(4))-160)/(35)xx10`
`=30+(120-106)/(35)xx10=30+(14)/(35)xx10`
`=30+4=34`
`D_(8)`=Size of `8((N)/(10))` item
= Size of `8((160)/(10))` th item
= Size of 128 th item
`D_(8)` lies in 14th cumulative frequency . the class interval corresponding to this cumulative frequency is 30-40.
Thus , `D_(8)=l_(1)+8(((N)/(10))-c.f.)/(f)xxi`
`=30+8((160/(10))-106)/(35)xx10=30+(128-106)/(35)xx10`
`=30+(22)/(35)xx10`
`=30+6.29`
`=36.29`
`P_(56)` = Size of `56((N)/(100))` th item
Size of 89.6th item
`P_(56)` lies in 106th cumulative frequency . the class interval corresponding to this cumulative frequency is 20-30.
Thus , `P_(56)=l_(1)+(56(N/(100))-c.f.)/(f)xxi`
`=20+(56(160/(100))-60)/(46)xx10`
`=20+(89.6-60)/(46)xx10`
`=20+(29.6)/(46)xx10`
`=20+6.43`
`=26.43`
Thus , `Q_(1)=14.74,Q_(3)=34,D_(8)=36.29,P_(56)=26.43.`