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The mean height of 200 students is 65 inches. The mean heights of boys and girls are 70 inches and 62 inches respectively and the standard deviations are 8 and 10 respectively. - Mathematics and Statistics

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Sum

The mean height of 200 students is 65 inches. The mean heights of boys and girls are 70 inches and 62 inches respectively and the standard deviations are 8 and 10 respectively. Find the number of boys and combined S.D.

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Solution

Let n1 and n2 be the number of boys and girls respectively.
Let n = 200, `bar("x"_"c")` = 65, `bar("x"_1)` = 70, `bar("x"_2)` = 62, σ1 = 8, σ2 = 10

Here, n1 + n2 = n
∴ n1 + n2 = 200  ............(i)
Combined mean is given by

`bar("x"_"c") = ("n"_1bar("x"_1) + "n"_2 bar("x"_2))/("n"_1 + "n"_2)`

∴ 65 = `("n"_1 (70) + "n"_2 (62))/200` ........[From (i)]

∴ 70n1 + 62n2 = 13000
∴ 35n1 + 31n2 = 6500  ..........(ii)
Solving (i) and (ii), we get 
n1 = 75, n2 = 125
Combined standard deviation is given by,

`sigma_"c" = sqrt(("n"_1(sigma_1^2 + "d"_1^2) + "n"_2 (sigma_2^2 + "d"_2^2))/("n"_1 + "n"_2)`

where d1 = `bar("x"_1) - bar("x"_"c")`, d2 = `bar("x"_2) - bar("x"_"c")`

∴ d1 = 70 – 65 = 5 and d2 = 62 – 65 = – 3

∴ `sigma_"c" = sqrt((75(64 + 25) + 125(100 + 9))/200)`

= `sqrt((6675 + 13625)/200)`

= `sqrt(20300/200`

= `sqrt(101.5)`
= 10.07

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 13 | Page 35
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