# Find the Variance 1/2 , 1/9 , 1/16

12 , 19 , 116
The mean of a set of numbers is the sum divided by the number of terms.
x&OverBar;=12+19+1163
Simplify the numerator.
To write 12 as a fraction with a common denominator, multiply by 99.
x&OverBar;=12⋅99+19+1163
To write 19 as a fraction with a common denominator, multiply by 22.
x&OverBar;=12⋅99+19⋅22+1163
Write each expression with a common denominator of 18, by multiplying each by an appropriate factor of 1.
Multiply 12 and 99.
x&OverBar;=92⋅9+19⋅22+1163
Multiply 2 by 9.
x&OverBar;=918+19⋅22+1163
Multiply 19 and 22.
x&OverBar;=918+29⋅2+1163
Multiply 9 by 2.
x&OverBar;=918+218+1163
x&OverBar;=918+218+1163
Combine the numerators over the common denominator.
x&OverBar;=9+218+1163
x&OverBar;=1118+1163
To write 1118 as a fraction with a common denominator, multiply by 88.
x&OverBar;=1118⋅88+1163
To write 116 as a fraction with a common denominator, multiply by 99.
x&OverBar;=1118⋅88+116⋅993
Write each expression with a common denominator of 144, by multiplying each by an appropriate factor of 1.
Multiply 1118 and 88.
x&OverBar;=11⋅818⋅8+116⋅993
Multiply 18 by 8.
x&OverBar;=11⋅8144+116⋅993
Multiply 116 and 99.
x&OverBar;=11⋅8144+916⋅93
Multiply 16 by 9.
x&OverBar;=11⋅8144+91443
x&OverBar;=11⋅8144+91443
Combine the numerators over the common denominator.
x&OverBar;=11⋅8+91443
Simplify the numerator.
Multiply 11 by 8.
x&OverBar;=88+91443
x&OverBar;=971443
x&OverBar;=971443
x&OverBar;=971443
Multiply the numerator by the reciprocal of the denominator.
x&OverBar;=97144⋅13
Multiply 97144⋅13.
Multiply 97144 and 13.
x&OverBar;=97144⋅3
Multiply 144 by 3.
x&OverBar;=97432
x&OverBar;=97432
Divide.
x&OverBar;=0.2245370‾
The mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.
x&OverBar;=0.2
Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.
s2=∑i=1n⁡(xi-xavg)2n-1
Set up the formula for variance for this set of numbers.
s=(12-0.2)2+(19-0.2)2+(116-0.2)23-1
Simplify the result.
Simplify the numerator.
To write -0.2 as a fraction with a common denominator, multiply by 22.
s=(12-0.2⋅22)2+(19-0.2)2+(116-0.2)23-1
Combine -0.2 and 22.
s=(12+-0.2⋅22)2+(19-0.2)2+(116-0.2)23-1
Combine the numerators over the common denominator.
s=(1-0.2⋅22)2+(19-0.2)2+(116-0.2)23-1
Simplify the numerator.
Multiply -0.2 by 2.
s=(1-0.42)2+(19-0.2)2+(116-0.2)23-1
Subtract 0.4 from 1.
s=(0.62)2+(19-0.2)2+(116-0.2)23-1
s=(0.62)2+(19-0.2)2+(116-0.2)23-1
Divide 0.6 by 2.
s=0.32+(19-0.2)2+(116-0.2)23-1
Raise 0.3 to the power of 2.
s=0.09+(19-0.2)2+(116-0.2)23-1
To write -0.2 as a fraction with a common denominator, multiply by 99.
s=0.09+(19-0.2⋅99)2+(116-0.2)23-1
Combine -0.2 and 99.
s=0.09+(19+-0.2⋅99)2+(116-0.2)23-1
Combine the numerators over the common denominator.
s=0.09+(1-0.2⋅99)2+(116-0.2)23-1
Simplify the numerator.
Multiply -0.2 by 9.
s=0.09+(1-1.89)2+(116-0.2)23-1
Subtract 1.8 from 1.
s=0.09+(-0.89)2+(116-0.2)23-1
s=0.09+(-0.89)2+(116-0.2)23-1
Divide -0.8 by 9.
s=0.09+(-0.08‾)2+(116-0.2)23-1
Raise -0.08‾ to the power of 2.
s=0.09+0.00790123+(116-0.2)23-1
To write -0.2 as a fraction with a common denominator, multiply by 1616.
s=0.09+0.00790123+(116-0.2⋅1616)23-1
Combine -0.2 and 1616.
s=0.09+0.00790123+(116+-0.2⋅1616)23-1
Combine the numerators over the common denominator.
s=0.09+0.00790123+(1-0.2⋅1616)23-1
Simplify the numerator.
Multiply -0.2 by 16.
s=0.09+0.00790123+(1-3.216)23-1
Subtract 3.2 from 1.
s=0.09+0.00790123+(-2.216)23-1
s=0.09+0.00790123+(-2.216)23-1
Divide -2.2 by 16.
s=0.09+0.00790123+(-0.1375)23-1
Raise -0.1375 to the power of 2.
s=0.09+0.00790123+0.018906253-1
s=0.09790123+0.018906253-1
s=0.116807483-1
s=0.116807483-1
Simplify the expression.
Subtract 1 from 3.
s=0.116807482
Divide 0.11680748 by 2.
s=0.05840374
s=0.05840374
s=0.05840374
Approximate the result.
s2≈0.0584
Find the Variance 1/2 , 1/9 , 1/16
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