For a certain bivariate data, following information is available. X Y Mean 13 17 S.D. 3 2 Size 3. 10 10 Obtain the combined standard deviation. - Mathematics and Statistics

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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`, σx = 3;  σy = 2, nx = 10,  ny = 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,

σc = `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`

∴ d1 = 13 – 15 = – 2 and d2 = 17 –15 = 2.

∴ σc = `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)`

Concept: Standard Deviation for Combined Data
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Chapter 2: Measures of Dispersion - Exercise 2.3 [Page 33]

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