Find the Variance 21 , 12 , 6 , 7 , 9

$21$ , $12$ , $6$ , $7$ , $9$

Step 1

The mean of a set of numbers is the sum divided by the number of terms.

$\overline{x} = \frac{21 + 12 + 6 + 7 + 9}{5}$

Step 2

Simplify the numerator.

Add $21$ and $12$ .

$\overline{x} = \frac{33 + 6 + 7 + 9}{5}$

Add $33$ and $6$ .

$\overline{x} = \frac{39 + 7 + 9}{5}$

Add $39$ and $7$ .

$\overline{x} = \frac{46 + 9}{5}$

Add $46$ and $9$ .

$\overline{x} = \frac{55}{5}$

Step 3

Divide $55$ by $5$ .

$\overline{x} = 11$

Step 4

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

$s^{2} = \sum_{i = 1}^{n} \frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}$

Step 5

Set up the formula for variance for this set of numbers.

$s = \frac{{(21 - 11)}^{2} + {(12 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $11$ from $21$ .

$s = \frac{10^{2} + {(12 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

Raise $10$ to the power of $2$ .

$s = \frac{100 + {(12 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

Subtract $11$ from $12$ .

$s = \frac{100 + 1^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

One to any power is one.

$s = \frac{100 + 1 + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

Subtract $11$ from $6$ .

$s = \frac{100 + 1 + {(- 5)}^{2} + {(7 - 11)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

Raise $- 5$ to the power of $2$ .

$s = \frac{100 + 1 + 25 + {(7 - 11)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

Subtract $11$ from $7$ .

$s = \frac{100 + 1 + 25 + {(- 4)}^{2} + {(9 - 11)}^{2}}{5 - 1}$

Raise $- 4$ to the power of $2$ .

$s = \frac{100 + 1 + 25 + 16 + {(9 - 11)}^{2}}{5 - 1}$

Subtract $11$ from $9$ .

$s = \frac{100 + 1 + 25 + 16 + {(- 2)}^{2}}{5 - 1}$

Raise $- 2$ to the power of $2$ .

$s = \frac{100 + 1 + 25 + 16 + 4}{5 - 1}$

Add $100$ and $1$ .

$s = \frac{101 + 25 + 16 + 4}{5 - 1}$

Add $101$ and $25$ .

$s = \frac{126 + 16 + 4}{5 - 1}$

Add $126$ and $16$ .

$s = \frac{142 + 4}{5 - 1}$

Add $142$ and $4$ .

$s = \frac{146}{5 - 1}$

Reduce the expression by cancelling the common factors.

Subtract $1$ from $5$ .

$s = \frac{146}{4}$

Cancel the common factor of $146$ and $4$ .

Factor $2$ out of $146$ .

$s = \frac{2 (73)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$s = \frac{2 \cdot 73}{2 \cdot 2}$

Cancel the common factor.

$s = \frac{2 \cdot 73}{2 \cdot 2}$

Rewrite the expression.

$s = \frac{73}{2}$

Step 7

Approximate the result.

$s^{2} \approx 36.5$

Find the Variance 21 , 12 , 6 , 7 , 9

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