# Find the Standard Deviation 6 , 6 , 10 , 8 , 10 , 8

6 , 6 , 10 , 8 , 10 , 8
Find the mean.
The mean of a set of numbers is the sum divided by the number of terms.
x&OverBar;=6+6+10+8+10+86
Cancel the common factor of 6+6+10+8+10+8 and 6.
Factor 2 out of 6.
x&OverBar;=2(3)+6+10+8+10+86
Factor 2 out of 6.
x&OverBar;=2⋅3+2⋅3+10+8+10+86
Factor 2 out of 2⋅3+2⋅3.
x&OverBar;=2⋅(3+3)+10+8+10+86
Factor 2 out of 10.
x&OverBar;=2⋅(3+3)+2(5)+8+10+86
Factor 2 out of 2⋅(3+3)+2(5).
x&OverBar;=2⋅(3+3+5)+8+10+86
Factor 2 out of 8.
x&OverBar;=2⋅(3+3+5)+2(4)+10+86
Factor 2 out of 2⋅(3+3+5)+2(4).
x&OverBar;=2⋅(3+3+5+4)+10+86
Factor 2 out of 10.
x&OverBar;=2⋅(3+3+5+4)+2(5)+86
Factor 2 out of 2⋅(3+3+5+4)+2(5).
x&OverBar;=2⋅(3+3+5+4+5)+86
Factor 2 out of 8.
x&OverBar;=2⋅(3+3+5+4+5)+2(4)6
Factor 2 out of 2⋅(3+3+5+4+5)+2(4).
x&OverBar;=2⋅(3+3+5+4+5+4)6
Cancel the common factors.
Factor 2 out of 6.
x&OverBar;=2⋅(3+3+5+4+5+4)2(3)
Cancel the common factor.
x&OverBar;=2⋅(3+3+5+4+5+4)2⋅3
Rewrite the expression.
x&OverBar;=3+3+5+4+5+43
x&OverBar;=3+3+5+4+5+43
x&OverBar;=3+3+5+4+5+43
Simplify the numerator.
x&OverBar;=6+5+4+5+43
x&OverBar;=11+4+5+43
x&OverBar;=15+5+43
x&OverBar;=20+43
x&OverBar;=243
x&OverBar;=243
Divide 24 by 3.
x&OverBar;=8
x&OverBar;=8
Simplify each value in the list.
Convert 6 to a decimal value.
6
Convert 10 to a decimal value.
10
Convert 8 to a decimal value.
8
Convert 10 to a decimal value.
10
Convert 8 to a decimal value.
8
The simplified values are 6,6,10,8,10,8.
6,6,10,8,10,8
6,6,10,8,10,8
Set up the formula for sample standard deviation. The standard deviation of a set of values is a measure of the spread of its values.
s=∑i=1n⁡(xi-xavg)2n-1
Set up the formula for standard deviation for this set of numbers.
s=(6-8)2+(6-8)2+(10-8)2+(8-8)2+(10-8)2+(8-8)26-1
Simplify the result.
Subtract 8 from 6.
s=(-2)2+(6-8)2+(10-8)2+(8-8)2+(10-8)2+(8-8)26-1
Raise -2 to the power of 2.
s=4+(6-8)2+(10-8)2+(8-8)2+(10-8)2+(8-8)26-1
Subtract 8 from 6.
s=4+(-2)2+(10-8)2+(8-8)2+(10-8)2+(8-8)26-1
Raise -2 to the power of 2.
s=4+4+(10-8)2+(8-8)2+(10-8)2+(8-8)26-1
Subtract 8 from 10.
s=4+4+22+(8-8)2+(10-8)2+(8-8)26-1
Raise 2 to the power of 2.
s=4+4+4+(8-8)2+(10-8)2+(8-8)26-1
Subtract 8 from 8.
s=4+4+4+02+(10-8)2+(8-8)26-1
Raising 0 to any positive power yields 0.
s=4+4+4+0+(10-8)2+(8-8)26-1
Subtract 8 from 10.
s=4+4+4+0+22+(8-8)26-1
Raise 2 to the power of 2.
s=4+4+4+0+4+(8-8)26-1
Subtract 8 from 8.
s=4+4+4+0+4+026-1
Raising 0 to any positive power yields 0.
s=4+4+4+0+4+06-1
s=4+4+4+0+46-1
s=8+4+0+46-1
s=12+0+46-1
s=12+46-1
s=166-1
Subtract 1 from 6.
s=165
Rewrite 165 as 165.
s=165
Simplify the numerator.
Rewrite 16 as 42.
s=425
Pull terms out from under the radical, assuming positive real numbers.
s=45
s=45
Multiply 45 by 55.
s=45⋅55
Combine and simplify the denominator.
Multiply 45 and 55.
s=4555
Raise 5 to the power of 1.
s=4555
Raise 5 to the power of 1.
s=4555
Use the power rule aman=am+n to combine exponents.
s=4551+1
s=4552
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
s=45(512)2
Apply the power rule and multiply exponents, (am)n=amn.
s=45512⋅2
Combine 12 and 2.
s=45522
Cancel the common factor of 2.
Cancel the common factor.
s=45522
Divide 1 by 1.
s=455
s=455
Evaluate the exponent.
s=455
s=455
s=455
s=455
The standard deviation 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.
1.8
Find the Standard Deviation 6 , 6 , 10 , 8 , 10 , 8

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