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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
Solution
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, nx = 5, ny = 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 dx = `bar"x" - bar("x"_"c")`, dy = `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
dx = 6 – 7 = – 1 and dy = 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)`
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