12 , 19 , 116

The mean of a set of numbers is the sum divided by the number of terms.

x‾=12+19+1163

To write 12 as a fraction with a common denominator, multiply by 99.

x‾=12⋅99+19+1163

To write 19 as a fraction with a common denominator, multiply by 22.

x‾=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‾=92⋅9+19⋅22+1163

Multiply 2 by 9.

x‾=918+19⋅22+1163

Multiply 19 and 22.

x‾=918+29⋅2+1163

Multiply 9 by 2.

x‾=918+218+1163

x‾=918+218+1163

Combine the numerators over the common denominator.

x‾=9+218+1163

Add 9 and 2.

x‾=1118+1163

To write 1118 as a fraction with a common denominator, multiply by 88.

x‾=1118⋅88+1163

To write 116 as a fraction with a common denominator, multiply by 99.

x‾=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‾=11⋅818⋅8+116⋅993

Multiply 18 by 8.

x‾=11⋅8144+116⋅993

Multiply 116 and 99.

x‾=11⋅8144+916⋅93

Multiply 16 by 9.

x‾=11⋅8144+91443

x‾=11⋅8144+91443

Combine the numerators over the common denominator.

x‾=11⋅8+91443

Simplify the numerator.

Multiply 11 by 8.

x‾=88+91443

Add 88 and 9.

x‾=971443

x‾=971443

x‾=971443

Multiply the numerator by the reciprocal of the denominator.

x‾=97144⋅13

Multiply 97144 and 13.

x‾=97144⋅3

Multiply 144 by 3.

x‾=97432

x‾=97432

Divide.

x‾=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‾=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 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

Add 0.09 and 0.00790123.

s=0.09790123+0.018906253-1

Add 0.09790123 and 0.01890625.

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