Find the Variance 10 , 12 , 14 , 15 , 18 , 20 , 23

$10$ , $12$ , $14$ , $15$ , $18$ , $20$ , $23$

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

$\overline{x} = \frac{10 + 12 + 14 + 15 + 18 + 20 + 23}{7}$

Simplify the numerator.

Add $10$ and $12$ .

$\overline{x} = \frac{22 + 14 + 15 + 18 + 20 + 23}{7}$

Add $22$ and $14$ .

$\overline{x} = \frac{36 + 15 + 18 + 20 + 23}{7}$

Add $36$ and $15$ .

$\overline{x} = \frac{51 + 18 + 20 + 23}{7}$

Add $51$ and $18$ .

$\overline{x} = \frac{69 + 20 + 23}{7}$

Add $69$ and $20$ .

$\overline{x} = \frac{89 + 23}{7}$

Add $89$ and $23$ .

$\overline{x} = \frac{112}{7}$

Divide $112$ by $7$ .

$\overline{x} = 16$

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

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

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

$s = \frac{{(10 - 16)}^{2} + {(12 - 16)}^{2} + {(14 - 16)}^{2} + {(15 - 16)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

Simplify the result.

Simplify the numerator.

Subtract $16$ from $10$ .

$s = \frac{{(- 6)}^{2} + {(12 - 16)}^{2} + {(14 - 16)}^{2} + {(15 - 16)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

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

$s = \frac{36 + {(12 - 16)}^{2} + {(14 - 16)}^{2} + {(15 - 16)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

Subtract $16$ from $12$ .

$s = \frac{36 + {(- 4)}^{2} + {(14 - 16)}^{2} + {(15 - 16)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

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

$s = \frac{36 + 16 + {(14 - 16)}^{2} + {(15 - 16)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

Subtract $16$ from $14$ .

$s = \frac{36 + 16 + {(- 2)}^{2} + {(15 - 16)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

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

$s = \frac{36 + 16 + 4 + {(15 - 16)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

Subtract $16$ from $15$ .

$s = \frac{36 + 16 + 4 + {(- 1)}^{2} + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

Raise $- 1$ to the power of $2$ .

$s = \frac{36 + 16 + 4 + 1 + {(18 - 16)}^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

Subtract $16$ from $18$ .

$s = \frac{36 + 16 + 4 + 1 + 2^{2} + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

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

$s = \frac{36 + 16 + 4 + 1 + 4 + {(20 - 16)}^{2} + {(23 - 16)}^{2}}{7 - 1}$

Subtract $16$ from $20$ .

$s = \frac{36 + 16 + 4 + 1 + 4 + 4^{2} + {(23 - 16)}^{2}}{7 - 1}$

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

$s = \frac{36 + 16 + 4 + 1 + 4 + 16 + {(23 - 16)}^{2}}{7 - 1}$

Subtract $16$ from $23$ .

$s = \frac{36 + 16 + 4 + 1 + 4 + 16 + 7^{2}}{7 - 1}$

Raise $7$ to the power of $2$ .

$s = \frac{36 + 16 + 4 + 1 + 4 + 16 + 49}{7 - 1}$

Add $36$ and $16$ .

$s = \frac{52 + 4 + 1 + 4 + 16 + 49}{7 - 1}$

Add $52$ and $4$ .

$s = \frac{56 + 1 + 4 + 16 + 49}{7 - 1}$

Add $56$ and $1$ .

$s = \frac{57 + 4 + 16 + 49}{7 - 1}$

Add $57$ and $4$ .

$s = \frac{61 + 16 + 49}{7 - 1}$

Add $61$ and $16$ .

$s = \frac{77 + 49}{7 - 1}$

Add $77$ and $49$ .

$s = \frac{126}{7 - 1}$

Simplify the expression.

Subtract $1$ from $7$ .

$s = \frac{126}{6}$

Divide $126$ by $6$ .

$s = 21$

Approximate the result.

$s^{2} \approx 21$

Find the Variance 10 , 12 , 14 , 15 , 18 , 20 , 23

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