# What is the variance of {9, 4, -5, 7, 12, -8}?

Jun 25, 2016

$\frac{1913}{30}$

#### Explanation:

Consider the set $\text{X}$ of numbers $9 , 4 , - 5 , 7 , 12 , - 8$

Step 1:

$\text{Mean" = "Sum of X values" /"N (Number of Values)}$

$= \frac{9 + 4 + \left(- 5\right) + 7 + 12 + \left(- 8\right)}{6}$

$= \frac{19}{6}$

Step 2:

In order to find the variance, subtract the mean from each of the values,

$9 - \frac{19}{6} = \frac{54}{6} - \frac{19}{6} = \frac{35}{6}$

$4 - \frac{19}{6} = \frac{24}{6} - \frac{19}{6} = \frac{5}{6}$

$- 5 - \frac{19}{6} = - \frac{30}{6} - \frac{19}{6} = - \frac{49}{6}$

$7 - \frac{19}{6} = \frac{42}{6} - \frac{19}{6} = \frac{23}{6}$

$12 - \frac{19}{6} = \frac{72}{6} - \frac{19}{6} = \frac{53}{6}$

$- 8 - \frac{19}{6} = - \frac{48}{6} - \frac{19}{6} = - \frac{67}{6}$

Step 3:

Now square all of the answers that you had gotten from subtraction.

${\left(\frac{35}{6}\right)}^{2} = \frac{1225}{36}$

${\left(\frac{5}{6}\right)}^{2} = \frac{25}{36}$

${\left(- \frac{49}{6}\right)}^{2} = \frac{2401}{36}$

${\left(\frac{23}{6}\right)}^{2} = \frac{529}{36}$

${\left(\frac{53}{6}\right)}^{2} = \frac{2809}{36}$

${\left(- \frac{67}{6}\right)}^{2} = \frac{4489}{36}$

Step 4:

Add all of the squared numbers,

$\frac{1225}{36} + \frac{25}{36} + \frac{2401}{36} + \frac{529}{36} + \frac{2809}{36} + \frac{4489}{36} = \frac{1913}{6}$

Step 5:

Divide the sum of squares by $\left(n - 1\right)$

$\frac{\frac{1913}{6}}{6 - 1} = \frac{\frac{1913}{6}}{5} = \frac{1913}{30} = 63.7 \left(6\right)$

Therefore

$\text{sample variance} = \frac{1913}{30}$

http://goodcalculators.com/standard-deviation-calculator/