Find the Variance 9 , 8 , 6 , 7 , 8 , 5 , 6 , 9 , 7 , 7

$9$ , $8$ , $6$ , $7$ , $8$ , $5$ , $6$ , $9$ , $7$ , $7$

Step 1

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

$\overline{x} = \frac{9 + 8 + 6 + 7 + 8 + 5 + 6 + 9 + 7 + 7}{10}$

Step 2

Simplify the numerator.

Add $9$ and $8$ .

$\overline{x} = \frac{17 + 6 + 7 + 8 + 5 + 6 + 9 + 7 + 7}{10}$

Add $17$ and $6$ .

$\overline{x} = \frac{23 + 7 + 8 + 5 + 6 + 9 + 7 + 7}{10}$

Add $23$ and $7$ .

$\overline{x} = \frac{30 + 8 + 5 + 6 + 9 + 7 + 7}{10}$

Add $30$ and $8$ .

$\overline{x} = \frac{38 + 5 + 6 + 9 + 7 + 7}{10}$

Add $38$ and $5$ .

$\overline{x} = \frac{43 + 6 + 9 + 7 + 7}{10}$

Add $43$ and $6$ .

$\overline{x} = \frac{49 + 9 + 7 + 7}{10}$

Add $49$ and $9$ .

$\overline{x} = \frac{58 + 7 + 7}{10}$

Add $58$ and $7$ .

$\overline{x} = \frac{65 + 7}{10}$

Add $65$ and $7$ .

$\overline{x} = \frac{72}{10}$

Step 3

Cancel the common factor of $72$ and $10$ .

Factor $2$ out of $72$ .

$\overline{x} = \frac{2 (36)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\overline{x} = \frac{2 \cdot 36}{2 \cdot 5}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 36}{2 \cdot 5}$

Rewrite the expression.

$\overline{x} = \frac{36}{5}$

Step 4

Divide.

$\overline{x} = 7.2$

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.

$s = \frac{{(9 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract $7.2$ from $9$ .

$s = \frac{{1.8}^{2} + {(8 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Raise $1.8$ to the power of $2$ .

$s = \frac{3.24 + {(8 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $8$ .

$s = \frac{3.24 + {0.8}^{2} + {(6 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + {(6 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $6$ .

$s = \frac{3.24 + 0.64 + {(- 1.2)}^{2} + {(7 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + 1.44 + {(7 - 7.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $7$ .

$s = \frac{3.24 + 0.64 + 1.44 + {(- 0.2)}^{2} + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + {(8 - 7.2)}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $8$ .

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + {0.8}^{2} + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + {(5 - 7.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $5$ .

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + {(- 2.2)}^{2} + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + {(6 - 7.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $6$ .

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + {(- 1.2)}^{2} + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + {(9 - 7.2)}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $9$ .

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + {1.8}^{2} + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Raise $1.8$ to the power of $2$ .

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + 3.24 + {(7 - 7.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $7$ .

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + 3.24 + {(- 0.2)}^{2} + {(7 - 7.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + 3.24 + 0.04 + {(7 - 7.2)}^{2}}{10 - 1}$

Subtract $7.2$ from $7$ .

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + 3.24 + 0.04 + {(- 0.2)}^{2}}{10 - 1}$

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

$s = \frac{3.24 + 0.64 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + 3.24 + 0.04 + 0.04}{10 - 1}$

Add $3.24$ and $0.64$ .

$s = \frac{3.88 + 1.44 + 0.04 + 0.64 + 4.84 + 1.44 + 3.24 + 0.04 + 0.04}{10 - 1}$

Add $3.88$ and $1.44$ .

$s = \frac{5.32 + 0.04 + 0.64 + 4.84 + 1.44 + 3.24 + 0.04 + 0.04}{10 - 1}$

Add $5.32$ and $0.04$ .

$s = \frac{5.36 + 0.64 + 4.84 + 1.44 + 3.24 + 0.04 + 0.04}{10 - 1}$

Add $5.36$ and $0.64$ .

$s = \frac{6 + 4.84 + 1.44 + 3.24 + 0.04 + 0.04}{10 - 1}$

Add $6$ and $4.84$ .

$s = \frac{10.84 + 1.44 + 3.24 + 0.04 + 0.04}{10 - 1}$

Add $10.84$ and $1.44$ .

$s = \frac{12.28 + 3.24 + 0.04 + 0.04}{10 - 1}$

Add $12.28$ and $3.24$ .

$s = \frac{15.52 + 0.04 + 0.04}{10 - 1}$

Add $15.52$ and $0.04$ .

$s = \frac{15.56 + 0.04}{10 - 1}$

Add $15.56$ and $0.04$ .

$s = \frac{15.6}{10 - 1}$

Simplify the expression.

Subtract $1$ from $10$ .

$s = \frac{15.6}{9}$

Divide $15.6$ by $9$ .

$s = 1.7\overline{3}$

Step 8

Approximate the result.

$s^{2} \approx 1.7333$

Find the Variance 9 , 8 , 6 , 7 , 8 , 5 , 6 , 9 , 7 , 7

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