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

Advertisement
Advertisement
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.

Advertisement

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
  Is there an error in this question or solution?
Chapter 2: Measures of Dispersion - Miscellaneous Exercise 2 [Page 35]

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
Share
Notifications



      Forgot password?
Use app×