Question

From the following data available for 5 pairs of observations of two variables x and y, obtain the combined S.D. for all 10 observations. 
where, `sum_("i" = 1)^"n" "x"_"i"` = 30, `sum_("i" = 1)^"n" "y"_"i"` = 40, `sum_("i" = 1)^"n" "x"_"i"^2` = 225, `sum_("i" = 1)^"n" "y"_"i"^2` = 340

Answer 1

Here, `sum_("i" = 1)^"n" "x"_"i"` = 30, `sum_("i" = 1)^"n" "y"_"i"` = 40, `sum_("i" = 1)^"n" "x"_"i"^2` = 225, `sum_("i" = 1)^"n" "y"_"i"^2` = 340, n_x = 5, n_y = 5

`bar"x" = (sum"x"_"i")/"n"_"x" = 30/5` = 6,

`bar"y" = (sum"y"_"i")/"n"_"y" = 40/5` = 8

Combined mean is given by 

`bar("x"_"c") = ("n"_"x" bar"x" + "n"_"y" bar"y")/("n"_"x" + "n"_"y")`

= `(5(6) + 5(8))/(5 + 5)`

= `(30 + 40)/10`

= `70/10`

= 7
Combined standard deviation is given by,

`sigma_"c" = sqrt(("n"_"x" (sigma_"x"^2 + "d"_"x"^2) + "n"_"y" (sigma_"y"^2 + "d"_"y"^2))/("n"_"x" + "n"_"y")`

Where d_x = `bar"x" - bar("x"_"c")`, d_y = `bar"y" - bar("x"_"c")`

`sigma_"x"^2 = 1/"n"_"x" sum"x"_"i"^2 - (bar"x")^2`

= `1/5(225) - (6)^2`

= 45 – 36
= 9
`sigma_"y"^2 = 1/"n"_"y" sum"y"_"i"^2 - (bar"y")^2`

= `1/5(340) - (8)^2`

= 68 – 64
= 4
d_x = 6 – 7 = – 1 and d_y = 8 – 7 = 1

∴ `sigma_"c" = sqrt((5[9 + (-1)^2] + 5[4 + (1)^2])/(5 + 5)`

= `sqrt((5(9 + 1) + 5(4 + 1))/10`

= `sqrt((5(10) + 5(5))/10`

= `sqrt((50 + 25)/10`

= `sqrt(75/10)`

= `sqrt(7.5)`