Find the Variance 18 , 14 , 4 , 7 , 11

$18$ , $14$ , $4$ , $7$ , $11$

Step 1

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

$\overline{x} = \frac{18 + 14 + 4 + 7 + 11}{5}$

Step 2

Simplify the numerator.

Add $18$ and $14$ .

$\overline{x} = \frac{32 + 4 + 7 + 11}{5}$

Add $32$ and $4$ .

$\overline{x} = \frac{36 + 7 + 11}{5}$

Add $36$ and $7$ .

$\overline{x} = \frac{43 + 11}{5}$

Add $43$ and $11$ .

$\overline{x} = \frac{54}{5}$

Step 3

Divide.

$\overline{x} = 10.8$

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.

$s = \frac{{(18 - 10.8)}^{2} + {(14 - 10.8)}^{2} + {(4 - 10.8)}^{2} + {(7 - 10.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $10.8$ from $18$ .

$s = \frac{{7.2}^{2} + {(14 - 10.8)}^{2} + {(4 - 10.8)}^{2} + {(7 - 10.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Raise $7.2$ to the power of $2$ .

$s = \frac{51.84 + {(14 - 10.8)}^{2} + {(4 - 10.8)}^{2} + {(7 - 10.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Subtract $10.8$ from $14$ .

$s = \frac{51.84 + {3.2}^{2} + {(4 - 10.8)}^{2} + {(7 - 10.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Raise $3.2$ to the power of $2$ .

$s = \frac{51.84 + 10.24 + {(4 - 10.8)}^{2} + {(7 - 10.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Subtract $10.8$ from $4$ .

$s = \frac{51.84 + 10.24 + {(- 6.8)}^{2} + {(7 - 10.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Raise $- 6.8$ to the power of $2$ .

$s = \frac{51.84 + 10.24 + 46.24 + {(7 - 10.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Subtract $10.8$ from $7$ .

$s = \frac{51.84 + 10.24 + 46.24 + {(- 3.8)}^{2} + {(11 - 10.8)}^{2}}{5 - 1}$

Raise $- 3.8$ to the power of $2$ .

$s = \frac{51.84 + 10.24 + 46.24 + 14.44 + {(11 - 10.8)}^{2}}{5 - 1}$

Subtract $10.8$ from $11$ .

$s = \frac{51.84 + 10.24 + 46.24 + 14.44 + {0.2}^{2}}{5 - 1}$

Raise $0.2$ to the power of $2$ .

$s = \frac{51.84 + 10.24 + 46.24 + 14.44 + 0.04}{5 - 1}$

Add $51.84$ and $10.24$ .

$s = \frac{62.08 + 46.24 + 14.44 + 0.04}{5 - 1}$

Add $62.08$ and $46.24$ .

$s = \frac{108.32 + 14.44 + 0.04}{5 - 1}$

Add $108.32$ and $14.44$ .

$s = \frac{122.76 + 0.04}{5 - 1}$

Add $122.76$ and $0.04$ .

$s = \frac{122.8}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$s = \frac{122.8}{4}$

Divide $122.8$ by $4$ .

$s = 30.7$

Step 7

Approximate the result.

$s^{2} \approx 30.7$

Find the Variance 18 , 14 , 4 , 7 , 11

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