Find the Variance 16 , 10 , 10 , 15 , 14

$16$ , $10$ , $10$ , $15$ , $14$

Step 1

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

$\overline{x} = \frac{16 + 10 + 10 + 15 + 14}{5}$

Step 2

Simplify the numerator.

Add $16$ and $10$ .

$\overline{x} = \frac{26 + 10 + 15 + 14}{5}$

Add $26$ and $10$ .

$\overline{x} = \frac{36 + 15 + 14}{5}$

Add $36$ and $15$ .

$\overline{x} = \frac{51 + 14}{5}$

Add $51$ and $14$ .

$\overline{x} = \frac{65}{5}$

Step 3

Divide $65$ by $5$ .

$\overline{x} = 13$

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.

$s = \frac{{(16 - 13)}^{2} + {(10 - 13)}^{2} + {(10 - 13)}^{2} + {(15 - 13)}^{2} + {(14 - 13)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $13$ from $16$ .

$s = \frac{3^{2} + {(10 - 13)}^{2} + {(10 - 13)}^{2} + {(15 - 13)}^{2} + {(14 - 13)}^{2}}{5 - 1}$

Raise $3$ to the power of $2$ .

$s = \frac{9 + {(10 - 13)}^{2} + {(10 - 13)}^{2} + {(15 - 13)}^{2} + {(14 - 13)}^{2}}{5 - 1}$

Subtract $13$ from $10$ .

$s = \frac{9 + {(- 3)}^{2} + {(10 - 13)}^{2} + {(15 - 13)}^{2} + {(14 - 13)}^{2}}{5 - 1}$

Raise $- 3$ to the power of $2$ .

$s = \frac{9 + 9 + {(10 - 13)}^{2} + {(15 - 13)}^{2} + {(14 - 13)}^{2}}{5 - 1}$

Subtract $13$ from $10$ .

$s = \frac{9 + 9 + {(- 3)}^{2} + {(15 - 13)}^{2} + {(14 - 13)}^{2}}{5 - 1}$

Raise $- 3$ to the power of $2$ .

$s = \frac{9 + 9 + 9 + {(15 - 13)}^{2} + {(14 - 13)}^{2}}{5 - 1}$

Subtract $13$ from $15$ .

$s = \frac{9 + 9 + 9 + 2^{2} + {(14 - 13)}^{2}}{5 - 1}$

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

$s = \frac{9 + 9 + 9 + 4 + {(14 - 13)}^{2}}{5 - 1}$

Subtract $13$ from $14$ .

$s = \frac{9 + 9 + 9 + 4 + 1^{2}}{5 - 1}$

One to any power is one.

$s = \frac{9 + 9 + 9 + 4 + 1}{5 - 1}$

Add $9$ and $9$ .

$s = \frac{18 + 9 + 4 + 1}{5 - 1}$

Add $18$ and $9$ .

$s = \frac{27 + 4 + 1}{5 - 1}$

Add $27$ and $4$ .

$s = \frac{31 + 1}{5 - 1}$

Add $31$ and $1$ .

$s = \frac{32}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$s = \frac{32}{4}$

Divide $32$ by $4$ .

$s = 8$

Step 7

Approximate the result.

$s^{2} \approx 8$

Find the Variance 16 , 10 , 10 , 15 , 14

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