Question

What is the variance of {-4, 5, -7, 0, -1, 10}?

Answer 1

Variance (`sigma_"pop"^2`) `= 31 7/12`

Explanation:

Population data:
`color(white)("XXX"){-4,5,-7,0,-1,10}`

Sum of population data:
`color(white)("XXX")(-4)+5+(-7)+0+(-1)+10=3`

Population size:
#color(white)("XXX")6

Mean:
`color(white)("XXX")3/6 = 1/2=0.5`

Deviations from Mean:
`color(white)("XXX"){(-4-0.5), (5-0.5), (-7-0.5), (0-0.5),(-1-0.5),(10-0.5)}`

`color(white)("XXX")= {-4.5,4.5,-7.5,-0.5,-1.5,9.5}`

Squares of Deviations from Mean:
`color(white)("XXX"){20.25,20.25,56.25,0.25,2.25,90.25}`

Sum of Squares of Deviations from Mean:
`color(white)("XXX")189.5`

Variance: `sigma_"pop"^2 = ("sum of squares of deviations from mean")/("population size")`
`color(white)("XXX")189.5/6 = 31 7/12 = 31.58bar3`

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Of course we would not normally do all these steps manually
(the above was strictly for educational purposes)

We would normally use a calculator or spreadsheet with a built in function as:

If you wanted the Sample Variance

• `sigma_"sample"^2 = ("sum of squares of deviations from mean")/("population size"-1)`
  and
• the (Excel) spreadsheet built-in function is VAR(A2:A7)