# 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

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.

#### 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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