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

Dec 9, 2015

Variance (${\sigma}_{\text{pop}}^{2}$) $= 31 \frac{7}{12}$

#### Explanation:

Population data:
$\textcolor{w h i t e}{\text{XXX}} \left\{- 4 , 5 , - 7 , 0 , - 1 , 10\right\}$

Sum of population data:
$\textcolor{w h i t e}{\text{XXX}} \left(- 4\right) + 5 + \left(- 7\right) + 0 + \left(- 1\right) + 10 = 3$

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

Mean:
$\textcolor{w h i t e}{\text{XXX}} \frac{3}{6} = \frac{1}{2} = 0.5$

Deviations from Mean:
$\textcolor{w h i t e}{\text{XXX}} \left\{\begin{matrix}- 4 - 0.5 \\ 5 - 0.5 \\ - 7 - 0.5 \\ 0 - 0.5 \\ - 1 - 0.5 \\ 10 - 0.5\end{matrix}\right\}$

$\textcolor{w h i t e}{\text{XXX}} = \left\{- 4.5 , 4.5 , - 7.5 , - 0.5 , - 1.5 , 9.5\right\}$

Squares of Deviations from Mean:
$\textcolor{w h i t e}{\text{XXX}} \left\{20.25 , 20.25 , 56.25 , 0.25 , 2.25 , 90.25\right\}$

Sum of Squares of Deviations from Mean:
$\textcolor{w h i t e}{\text{XXX}} 189.5$

Variance: sigma_"pop"^2 = ("sum of squares of deviations from mean")/("population size")
$\textcolor{w h i t e}{\text{XXX}} \frac{189.5}{6} = 31 \frac{7}{12} = 31.58 \overline{3}$

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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)