Find the Variance 7 , 15 , 11 , 1 , 11

$7$ , $15$ , $11$ , $1$ , $11$

Step 1

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

$\overline{x} = \frac{7 + 15 + 11 + 1 + 11}{5}$

Step 2

Simplify the numerator.

Add $7$ and $15$ .

$\overline{x} = \frac{22 + 11 + 1 + 11}{5}$

Add $22$ and $11$ .

$\overline{x} = \frac{33 + 1 + 11}{5}$

Add $33$ and $1$ .

$\overline{x} = \frac{34 + 11}{5}$

Add $34$ and $11$ .

$\overline{x} = \frac{45}{5}$

Step 3

Divide $45$ by $5$ .

$\overline{x} = 9$

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.

$s = \frac{{(7 - 9)}^{2} + {(15 - 9)}^{2} + {(11 - 9)}^{2} + {(1 - 9)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $9$ from $7$ .

$s = \frac{{(- 2)}^{2} + {(15 - 9)}^{2} + {(11 - 9)}^{2} + {(1 - 9)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

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

$s = \frac{4 + {(15 - 9)}^{2} + {(11 - 9)}^{2} + {(1 - 9)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

Subtract $9$ from $15$ .

$s = \frac{4 + 6^{2} + {(11 - 9)}^{2} + {(1 - 9)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

Raise $6$ to the power of $2$ .

$s = \frac{4 + 36 + {(11 - 9)}^{2} + {(1 - 9)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

Subtract $9$ from $11$ .

$s = \frac{4 + 36 + 2^{2} + {(1 - 9)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

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

$s = \frac{4 + 36 + 4 + {(1 - 9)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

Subtract $9$ from $1$ .

$s = \frac{4 + 36 + 4 + {(- 8)}^{2} + {(11 - 9)}^{2}}{5 - 1}$

Raise $- 8$ to the power of $2$ .

$s = \frac{4 + 36 + 4 + 64 + {(11 - 9)}^{2}}{5 - 1}$

Subtract $9$ from $11$ .

$s = \frac{4 + 36 + 4 + 64 + 2^{2}}{5 - 1}$

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

$s = \frac{4 + 36 + 4 + 64 + 4}{5 - 1}$

Add $4$ and $36$ .

$s = \frac{40 + 4 + 64 + 4}{5 - 1}$

Add $40$ and $4$ .

$s = \frac{44 + 64 + 4}{5 - 1}$

Add $44$ and $64$ .

$s = \frac{108 + 4}{5 - 1}$

Add $108$ and $4$ .

$s = \frac{112}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$s = \frac{112}{4}$

Divide $112$ by $4$ .

$s = 28$

Step 7

Approximate the result.

$s^{2} \approx 28$

Find the Variance 7 , 15 , 11 , 1 , 11

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