Find the Variance 12 , 16 , 23 , 14 , 20 , 18 , 22 , 18 , 14 , 15

$12$ , $16$ , $23$ , $14$ , $20$ , $18$ , $22$ , $18$ , $14$ , $15$

Step 1

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

$\overline{x} = \frac{12 + 16 + 23 + 14 + 20 + 18 + 22 + 18 + 14 + 15}{10}$

Step 2

Simplify the numerator.

Add $12$ and $16$ .

$\overline{x} = \frac{28 + 23 + 14 + 20 + 18 + 22 + 18 + 14 + 15}{10}$

Add $28$ and $23$ .

$\overline{x} = \frac{51 + 14 + 20 + 18 + 22 + 18 + 14 + 15}{10}$

Add $51$ and $14$ .

$\overline{x} = \frac{65 + 20 + 18 + 22 + 18 + 14 + 15}{10}$

Add $65$ and $20$ .

$\overline{x} = \frac{85 + 18 + 22 + 18 + 14 + 15}{10}$

Add $85$ and $18$ .

$\overline{x} = \frac{103 + 22 + 18 + 14 + 15}{10}$

Add $103$ and $22$ .

$\overline{x} = \frac{125 + 18 + 14 + 15}{10}$

Add $125$ and $18$ .

$\overline{x} = \frac{143 + 14 + 15}{10}$

Add $143$ and $14$ .

$\overline{x} = \frac{157 + 15}{10}$

Add $157$ and $15$ .

$\overline{x} = \frac{172}{10}$

Step 3

Cancel the common factor of $172$ and $10$ .

Factor $2$ out of $172$ .

$\overline{x} = \frac{2 (86)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\overline{x} = \frac{2 \cdot 86}{2 \cdot 5}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 86}{2 \cdot 5}$

Rewrite the expression.

$\overline{x} = \frac{86}{5}$

Step 4

Divide.

$\overline{x} = 17.2$

Step 5

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 6

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

$s = \frac{{(12 - 17.2)}^{2} + {(16 - 17.2)}^{2} + {(23 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract $17.2$ from $12$ .

$s = \frac{{(- 5.2)}^{2} + {(16 - 17.2)}^{2} + {(23 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Raise $- 5.2$ to the power of $2$ .

$s = \frac{27.04 + {(16 - 17.2)}^{2} + {(23 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $16$ .

$s = \frac{27.04 + {(- 1.2)}^{2} + {(23 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Raise $- 1.2$ to the power of $2$ .

$s = \frac{27.04 + 1.44 + {(23 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $23$ .

$s = \frac{27.04 + 1.44 + {5.8}^{2} + {(14 - 17.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Raise $5.8$ to the power of $2$ .

$s = \frac{27.04 + 1.44 + 33.64 + {(14 - 17.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $14$ .

$s = \frac{27.04 + 1.44 + 33.64 + {(- 3.2)}^{2} + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

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

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + {(20 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $20$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + {2.8}^{2} + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Raise $2.8$ to the power of $2$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + {(18 - 17.2)}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $18$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + {0.8}^{2} + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Raise $0.8$ to the power of $2$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + {(22 - 17.2)}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $22$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + {4.8}^{2} + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Raise $4.8$ to the power of $2$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + {(18 - 17.2)}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $18$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + {0.8}^{2} + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Raise $0.8$ to the power of $2$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + 0.64 + {(14 - 17.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $14$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + 0.64 + {(- 3.2)}^{2} + {(15 - 17.2)}^{2}}{10 - 1}$

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

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + 0.64 + 10.24 + {(15 - 17.2)}^{2}}{10 - 1}$

Subtract $17.2$ from $15$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + 0.64 + 10.24 + {(- 2.2)}^{2}}{10 - 1}$

Raise $- 2.2$ to the power of $2$ .

$s = \frac{27.04 + 1.44 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + 0.64 + 10.24 + 4.84}{10 - 1}$

Add $27.04$ and $1.44$ .

$s = \frac{28.48 + 33.64 + 10.24 + 7.84 + 0.64 + 23.04 + 0.64 + 10.24 + 4.84}{10 - 1}$

Add $28.48$ and $33.64$ .

$s = \frac{62.12 + 10.24 + 7.84 + 0.64 + 23.04 + 0.64 + 10.24 + 4.84}{10 - 1}$

Add $62.12$ and $10.24$ .

$s = \frac{72.36 + 7.84 + 0.64 + 23.04 + 0.64 + 10.24 + 4.84}{10 - 1}$

Add $72.36$ and $7.84$ .

$s = \frac{80.2 + 0.64 + 23.04 + 0.64 + 10.24 + 4.84}{10 - 1}$

Add $80.2$ and $0.64$ .

$s = \frac{80.84 + 23.04 + 0.64 + 10.24 + 4.84}{10 - 1}$

Add $80.84$ and $23.04$ .

$s = \frac{103.88 + 0.64 + 10.24 + 4.84}{10 - 1}$

Add $103.88$ and $0.64$ .

$s = \frac{104.52 + 10.24 + 4.84}{10 - 1}$

Add $104.52$ and $10.24$ .

$s = \frac{114.76 + 4.84}{10 - 1}$

Add $114.76$ and $4.84$ .

$s = \frac{119.6}{10 - 1}$

Simplify the expression.

Subtract $1$ from $10$ .

$s = \frac{119.6}{9}$

Divide $119.6$ by $9$ .

$s = 13.2\overline{8}$

Step 8

Approximate the result.

$s^{2} \approx 13.2889$

Find the Variance 12 , 16 , 23 , 14 , 20 , 18 , 22 , 18 , 14 , 15

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