# 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. - Mathematics and Statistics

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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)

Concept: Standard Deviation for Combined Data
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board
Chapter 2 Measures of Dispersion
Miscellaneous Exercise 2 | Q 14 | Page 35