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Find the Variance 10 , 11 , 12 , 12 , 13 , 13 , 13 , 14 , 14 , 14 , 15 , 15 , 15 , 15 , 15 , 16 , 16 , 16 , 17 , 17 , 17 , 17 , 18 , 18 , 25

10 , 11 , 12 , 12 , 13 , 13 , 13 , 14 , 14 , 14 , 15 , 15 , 15 , 15 , 15 , 16 , 16 , 16 , 17 , 17 , 17 , 17 , 18 , 18 , 25
Step 1
The mean of a set of numbers is the sum divided by the number of terms.
‾x=10+11+12+12+13+13+13+14+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525

Step 2
Simplify the numerator.

Add 10 and 11.
‾x=21+12+12+13+13+13+14+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 21 and 12.
‾x=33+12+13+13+13+14+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 33 and 12.
‾x=45+13+13+13+14+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 45 and 13.
‾x=58+13+13+14+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 58 and 13.
‾x=71+13+14+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 71 and 13.
‾x=84+14+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 84 and 14.
‾x=98+14+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 98 and 14.
‾x=112+14+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 112 and 14.
‾x=126+15+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 126 and 15.
‾x=141+15+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 141 and 15.
‾x=156+15+15+15+16+16+16+17+17+17+17+18+18+2525
Add 156 and 15.
‾x=171+15+15+16+16+16+17+17+17+17+18+18+2525
Add 171 and 15.
‾x=186+15+16+16+16+17+17+17+17+18+18+2525
Add 186 and 15.
‾x=201+16+16+16+17+17+17+17+18+18+2525
Add 201 and 16.
‾x=217+16+16+17+17+17+17+18+18+2525
Add 217 and 16.
‾x=233+16+17+17+17+17+18+18+2525
Add 233 and 16.
‾x=249+17+17+17+17+18+18+2525
Add 249 and 17.
‾x=266+17+17+17+18+18+2525
Add 266 and 17.
‾x=283+17+17+18+18+2525
Add 283 and 17.
‾x=300+17+18+18+2525
Add 300 and 17.
‾x=317+18+18+2525
Add 317 and 18.
‾x=335+18+2525
Add 335 and 18.
‾x=353+2525
Add 353 and 25.
‾x=37825
‾x=37825

Step 3
Divide.
‾x=15.12

Step 4
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=15.1

Step 5
Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.
s2=n∑i=1(xi-xavg)2n-1

Step 6
Set up the formula for variance for this set of numbers.
s=(10-15.1)2+(11-15.1)2+(12-15.1)2+(12-15.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1

Step 7
Simplify the result.

Simplify the numerator.

Subtract 15.1 from 10.
s=(-5.1)2+(11-15.1)2+(12-15.1)2+(12-15.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -5.1 to the power of 2.
s=26.01+(11-15.1)2+(12-15.1)2+(12-15.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 11.
s=26.01+(-4.1)2+(12-15.1)2+(12-15.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -4.1 to the power of 2.
s=26.01+16.81+(12-15.1)2+(12-15.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 12.
s=26.01+16.81+(-3.1)2+(12-15.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -3.1 to the power of 2.
s=26.01+16.81+9.61+(12-15.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 12.
s=26.01+16.81+9.61+(-3.1)2+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -3.1 to the power of 2.
s=26.01+16.81+9.61+9.61+(13-15.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 13.
s=26.01+16.81+9.61+9.61+(-2.1)2+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -2.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+(13-15.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 13.
s=26.01+16.81+9.61+9.61+4.41+(-2.1)2+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -2.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+(13-15.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 13.
s=26.01+16.81+9.61+9.61+4.41+4.41+(-2.1)2+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -2.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+(14-15.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 14.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+(-1.1)2+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -1.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+(14-15.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 14.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+(-1.1)2+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -1.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+(14-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 14.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+(-1.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -1.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 15.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+(-0.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -0.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+(15-15.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 15.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+(-0.1)2+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -0.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+(15-15.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 15.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+(-0.1)2+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -0.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+0.01+(15-15.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 15.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+0.01+(-0.1)2+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -0.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+0.01+0.01+(15-15.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 15.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+0.01+0.01+(-0.1)2+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise -0.1 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+0.01+0.01+0.01+(16-15.1)2+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract 15.1 from 16.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+0.01+0.01+0.01+0.92+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Raise 0.9 to the power of 2.
s=26.01+16.81+9.61+9.61+4.41+4.41+4.41+1.21+1.21+1.21+0.01+0.01+0.01+0.01+0.01+0.81+(16-15.1)2+(16-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(17-15.1)2+(18-15.1)2+(18-15.1)2+(25-15.1)225-1
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Subtract from .
Raise to the power of .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Add and .
Simplify the expression.

Subtract from .
Divide by .

Step 8
Approximate the result.


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