1. Find Standard deviation (Method-1)
`N_1=40,bar X_1=10,sigma_1=1`
`N_2=60,bar X_2=15,sigma_2=2`
Solution:
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
`d_1 = bar x_1 - bar x_(12) = 10 - 13 = -3`
`d_2 = bar x_2 - bar x_(12) = 15 - 13 = 2`
Combined Standard deviation :
`sigma_(12) = sqrt(( N_1 * (sigma_1^2 + d_1^2) + N_2 * (sigma_2^2 + d_2^2) )/( N_1 + N_2 ))`
`sigma_(12) = sqrt(( 40 xx ( 1 + 9 ) + 60 xx ( 4 + 4 ) )/( 40 + 60 ))`
`sigma_(12) = sqrt(( 400 + 480 )/(100))`
`sigma_(12) = sqrt(880/100)`
`sigma_(12) = 2.9665`
Combined SD `=2.9665`
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`
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
`d_1=bar x_1-bar x_(12)=10-13=-3`
`d_2=bar x_2-bar x_(12)=15-13=2`
Combined Standard deviation :
`sigma_(12)=sqrt((N_1*(sigma_1^2+d_1^2)+N_2*(sigma_2^2+d_2^2))/(N_1+N_2))`
`sigma_(12)=sqrt((40 xx (1+9)+60 xx (4+4))/(40+60))`
`sigma_(12)=sqrt((400+480)/(100))`
`sigma_(12)=sqrt(880/100)`
`sigma_(12)=2.9665`
Combined SD `=2.9665`
This material is intended as a summary. Use your textbook for detail explanation.
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