1. Find Standard deviation (Method-2)
`N_1=40,bar X_1=10,sigma_1=1`
`N_2=60,bar X_2=15,sigma_2=2`
Solution:
First for Group(1) and Group(2)
`N_1=40,bar X_1=10,sigma_1=1`
`N_2=60,bar X_2=15,sigma_2=2`
Combined Mean :
`bar x_12 = ( N_1 * bar (x)_1 + N_2 * bar (x)_2 )/( N_1 + N_2 )`
`bar x_12 = ( 40 xx 10 + 60 xx 15 )/( 40 + 60 )`
`bar x_12 = ( 400 + 900 )/(100)`
`bar x_12 = 1300/100`
`bar x_12 = 13`
Combined Mean = 13
Combined Standard deviation :
`sigma_(12) = sqrt(((N_1 - 1) * sigma_1^2 + (N_2 - 1) * sigma_2^2 + (N_1 * N_2)/(N_1 + N_2) * (bar x_1^2 + bar x_2^2 - 2 bar x_1 bar x_2))/(N_1 + N_2 - 1))`
`sigma_(12) = sqrt(((40-1) xx 1^2 + (60-1) xx 2^2 + (40 xx 60)/(40 + 60) xx (10^2 + 15^2 - 2 xx 10 xx 15))/(40+60-1))`
`sigma_(12) = sqrt((39 xx 1 + 59 xx 4 + 2400/100 xx (100 + 225 - 300))/(99))`
`sigma_(12) = sqrt(875/99)`
`sigma_(12) = 2.9729`
2. Find Standard deviation from the following data
`N_1=40,sum X_1=400,V_1=1`
`N_2=60,sum X_2=900,V_2=4`
Solution:
Mean `bar x_1 = (sum x_1)/n``=400/40``=10`
Mean `bar x_2 = (sum x_2)/n``=900/60``=15`
`sigma_1=sqrt(text{Variance}_1)=sqrt(1)=1`
`sigma_2=sqrt(text{Variance}_2)=sqrt(4)=2`
First for Group(1) and Group(2)
`N_1=40,bar X_1=10,sigma_1=1`
`N_2=60,bar X_2=15,sigma_2=2`
Combined Mean :
`bar x_12 = (N_1 * bar (x)_1+N_2 * bar (x)_2)/(N_1+N_2)`
`bar x_12 = (40 xx 10+60 xx 15)/(40+60)`
`bar x_12 = (400+900)/(100)`
`bar x_12 = 1300/100`
`bar x_12 = 13`
Combined Mean = 13
Combined Standard deviation :
`sigma_(12)=sqrt(((N_1 - 1) * sigma_1^2+(N_2 - 1) * sigma_2^2+(N_1 * N_2)/(N_1 + N_2) * (bar x_1^2 + bar x_2^2 - 2 bar x_1 bar x_2))/(N_1 + N_2 - 1))`
`sigma_(12)=sqrt(((40-1) xx 1^2+(60-1) xx 2^2+(40 xx 60)/(40 + 60) xx (10^2 + 15^2 - 2 xx 10 xx 15))/(40+60-1))`
`sigma_(12)=sqrt((39 xx 1+59 xx 4+2400/100 xx (100 + 225 - 300))/(99))`
`sigma_(12)=sqrt(875/99)`
`sigma_(12)=2.9729`
This material is intended as a summary. Use your textbook for detail explanation.
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