Guided Example #3 - Variance

Find the variance.
35, 15, 20, 15, 35, 45, 40, 50, 60, 25

A. $15.24$ B. $223.22$ C. $49, 827.17$ D. $3.90$

Step 1

Find the sum of the values:
$Σ x = 15 + 15 + 20 + 25 + 35 + 35 + 40 + 45 + 50 + 60 = 340$

Step 2

Square each value, and then find their sum:
$Σ x^{2} = 15^{2} + 15^{2} + 20^{2} + 25^{2} + 35^{2} + 35^{2} + 40^{2} + 45^{2} + 50^{2} + 60^{2} = 13650$

Step 3

Substitute in $s^{2} = \frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1}$ , and solve.
$s^{2} = \frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1} = \frac{13650 - \frac{{(340)}^{2}}{10}}{9} = 223.22$

Step 4

Therefore the variance is $223.22$ .
The standard deviation is $s = \sqrt{223.22} = 15.24$ .

All Steps

Find the sum of the values:
$Σ x = 15 + 15 + 20 + 25 + 35 + 35 + 40 + 45 + 50 + 60 = 340$

Square each value, and then find their sum:
$Σ x^{2} = 15^{2} + 15^{2} + 20^{2} + 25^{2} + 35^{2} + 35^{2} + 40^{2} + 45^{2} + 50^{2} + 60^{2} = 13650$

Substitute in $s^{2} = \frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1}$ , and solve.
$s^{2} = \frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1} = \frac{13650 - \frac{{(340)}^{2}}{10}}{9} = 223.22$

Therefore the variance is $223.22$ .
The standard deviation is $s = \sqrt{223.22} = 15.24$ .