Question

The means of two samples of sizes 60 and 120 respectively are 35.4 and 30.9 and the standard deviations are 4 and 5. Obtain the standard deviation of the sample of size 180 obtained by combining the two samples.

Options

• 5.15

• 26.5

• 32.4

• 51.5

Answer 1

5.15

Explanation:

Let n₁ = 60, n₂ = 120, `barx_1` = 35.4, `barx_2` = 30.9, σ₁ = 4, σ₂ = 5

Combined mean `(barx_e)`

`=("n"_1barx_1 + "n"_2barx_2)/("n"_1 + "n"_2)`

`= (60 xx 35.4 + 120 xx 30.9)/(60 + 120)`

`= (2124 + 3708)/180`

`= 5832/180`

= 32.4

now, d₁ = `barx_1 - barx_e = 35.4 - 32.4 = 3`

d₂ =  `barx_2 - barx_e` = 30.9 - 32.4 = - 1.5

∴ Combined standard deviation (σ_e)

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

`= sqrt((60(4^2 + 3^2) + 120 [5^2 + (- 1.5)^2])/(60 + 120))`

`= sqrt((60(16 + 9) + 120(25 + 2.25))/(180))`

`= sqrt((60(25) + 120(27.25))/180)`

`= sqrt((1500 + 3270)/(180))`

= `sqrt(4770/180)`

`= sqrt26.5`

= 5.15