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.

#### Solution

Let n_{1} and n_{2} be the number of boys and girls respectively.

Let n = 200, `bar("x"_"c")` = 65, `bar("x"_1)` = 70, `bar("x"_2)` = 62, σ_{1} = 8, σ_{2 }= 10

Here, n_{1} + n_{2} = n

∴ n_{1} + n_{2} = 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)]

∴ 70n_{1} + 62n_{2} = 13000

∴ 35n_{1} + 31n_{2} = 6500 ..........(ii)

Solving (i) and (ii), we get

n_{1} = 75, n_{2 }= 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 d_{1} = `bar("x"_1) - bar("x"_"c")`, d_{2} = `bar("x"_2) - bar("x"_"c")`

∴ d_{1} = 70 – 65 = 5 and d_{2} = 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