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For certain bivariate data, the following information is available.
X | Y | |
Mean | 13 | 17 |
S.D. | 3 | 2 |
Size | 10 | 10 |
Obtain the combined standard deviation.
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Solution
`barx = 13; bary = 17`, σ_{x} = 3; σ_{y} = 2, n_{x} = 10, n_{y} = 10.
Combined Mean,
`bar(x_c) = ("n"_x barx + "n"_y bary)/("n"_x + "n"_y)`
= `(10(13) + (10)(17))/(10 + 10)`
= `(130 + 170)/(20)`
= `(300)/(20)`
∴ `bar(x_c)` = 15
Combined standard deviation is given by,
σ_{c} = `sqrt((n_x(σ_x^2 + d_x^2) + n_y(σ_y^2 + d_y^2))/(n_x + n_y)`
Where, `d_1 = barx - barx_c, d_2 = bary - barx_c`
∴ d_{1} = 13 – 15 = – 2 and d_{2} = 17 –15 = 2.
∴ σ_{c} = `sqrt((10[3^2 + (-2)^2] + 10(2^2 + 2^2))/(10 + 10)`
= `sqrt((10[9 + 4] + 10(4 + 4))/(20)`
= `sqrt((10(13) + 10(8))/(20)`
= `sqrt((130 + 80)/(20))`
= `sqrt((210)/(20))`
= `sqrt(10.5)`
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