# Find the Standard Deviation 20 , 40 , 16 , 22 , 35 , 23 , 23 , 29 , 28 , 38 20 , 40 , 16 , 22 , 35 , 23 , 23 , 29 , 28 , 38
Find the mean.
The mean of a set of numbers is the sum divided by the number of terms.
x&OverBar;=20+40+16+22+35+23+23+29+28+3810
Simplify the numerator.
Add 20 and 40.
x&OverBar;=60+16+22+35+23+23+29+28+3810
Add 60 and 16.
x&OverBar;=76+22+35+23+23+29+28+3810
Add 76 and 22.
x&OverBar;=98+35+23+23+29+28+3810
Add 98 and 35.
x&OverBar;=133+23+23+29+28+3810
Add 133 and 23.
x&OverBar;=156+23+29+28+3810
Add 156 and 23.
x&OverBar;=179+29+28+3810
Add 179 and 29.
x&OverBar;=208+28+3810
Add 208 and 28.
x&OverBar;=236+3810
Add 236 and 38.
x&OverBar;=27410
x&OverBar;=27410
Cancel the common factor of 274 and 10.
Factor 2 out of 274.
x&OverBar;=2(137)10
Cancel the common factors.
Factor 2 out of 10.
x&OverBar;=2⋅1372⋅5
Cancel the common factor.
x&OverBar;=2⋅1372⋅5
Rewrite the expression.
x&OverBar;=1375
x&OverBar;=1375
x&OverBar;=1375
Divide.
x&OverBar;=27.4
x&OverBar;=27.4
Simplify each value in the list.
Convert 20 to a decimal value.
20
Convert 40 to a decimal value.
40
Convert 16 to a decimal value.
16
Convert 22 to a decimal value.
22
Convert 35 to a decimal value.
35
Convert 23 to a decimal value.
23
Convert 29 to a decimal value.
29
Convert 28 to a decimal value.
28
Convert 38 to a decimal value.
38
The simplified values are 20,40,16,22,35,23,23,29,28,38.
20,40,16,22,35,23,23,29,28,38
20,40,16,22,35,23,23,29,28,38
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=(20-27.4)2+(40-27.4)2+(16-27.4)2+(22-27.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Simplify the result.
Subtract 27.4 from 20.
s=(-7.4)2+(40-27.4)2+(16-27.4)2+(22-27.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Raise -7.4 to the power of 2.
s=54.76+(40-27.4)2+(16-27.4)2+(22-27.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 40.
s=54.76+12.62+(16-27.4)2+(22-27.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Raise 12.6 to the power of 2.
s=54.76+158.76+(16-27.4)2+(22-27.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 16.
s=54.76+158.76+(-11.4)2+(22-27.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Raise -11.4 to the power of 2.
s=54.76+158.76+129.96+(22-27.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 22.
s=54.76+158.76+129.96+(-5.4)2+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Raise -5.4 to the power of 2.
s=54.76+158.76+129.96+29.16+(35-27.4)2+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 35.
s=54.76+158.76+129.96+29.16+7.62+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Raise 7.6 to the power of 2.
s=54.76+158.76+129.96+29.16+57.76+(23-27.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 23.
s=54.76+158.76+129.96+29.16+57.76+(-4.4)2+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Raise -4.4 to the power of 2.
s=54.76+158.76+129.96+29.16+57.76+19.36+(23-27.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 23.
s=54.76+158.76+129.96+29.16+57.76+19.36+(-4.4)2+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Raise -4.4 to the power of 2.
s=54.76+158.76+129.96+29.16+57.76+19.36+19.36+(29-27.4)2+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 29.
s=54.76+158.76+129.96+29.16+57.76+19.36+19.36+1.62+(28-27.4)2+(38-27.4)210-1
Raise 1.6 to the power of 2.
s=54.76+158.76+129.96+29.16+57.76+19.36+19.36+2.56+(28-27.4)2+(38-27.4)210-1
Subtract 27.4 from 28.
s=54.76+158.76+129.96+29.16+57.76+19.36+19.36+2.56+0.62+(38-27.4)210-1
Raise 0.6 to the power of 2.
s=54.76+158.76+129.96+29.16+57.76+19.36+19.36+2.56+0.36+(38-27.4)210-1
Subtract 27.4 from 38.
s=54.76+158.76+129.96+29.16+57.76+19.36+19.36+2.56+0.36+10.6210-1
Raise 10.6 to the power of 2.
s=54.76+158.76+129.96+29.16+57.76+19.36+19.36+2.56+0.36+112.3610-1
Add 54.76 and 158.76.
s=213.52+129.96+29.16+57.76+19.36+19.36+2.56+0.36+112.3610-1
Add 213.52 and 129.96.
s=343.48+29.16+57.76+19.36+19.36+2.56+0.36+112.3610-1
Add 343.48 and 29.16.
s=372.64+57.76+19.36+19.36+2.56+0.36+112.3610-1
Add 372.64 and 57.76.
s=430.4+19.36+19.36+2.56+0.36+112.3610-1
Add 430.4 and 19.36.
s=449.76+19.36+2.56+0.36+112.3610-1
Add 449.76 and 19.36.
s=469.12+2.56+0.36+112.3610-1
Add 469.12 and 2.56.
s=471.68+0.36+112.3610-1
Add 471.68 and 0.36.
s=472.04+112.3610-1
Add 472.04 and 112.36.
s=584.410-1
Subtract 1 from 10.
s=584.49
Divide 584.4 by 9.
s=64.93‾
s=64.93‾
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.
8.1
Find the Standard Deviation 20 , 40 , 16 , 22 , 35 , 23 , 23 , 29 , 28 , 38

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