Basic Math Examples

$7$ , $2$ , $6$ , $- 4$ , $- 4$ , $3$ , $0$ , $18$ , $- 1$

Step 1

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

$\overline{x} = \frac{7 + 2 + 6 - 4 - 4 + 3 + 0 + 18 - 1}{9}$

Step 2

Simplify the numerator.

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Step 2.1

Add $7$ and $2$ .

$\overline{x} = \frac{9 + 6 - 4 - 4 + 3 + 0 + 18 - 1}{9}$

Step 2.2

Add $9$ and $6$ .

$\overline{x} = \frac{15 - 4 - 4 + 3 + 0 + 18 - 1}{9}$

Step 2.3

Subtract $4$ from $15$ .

$\overline{x} = \frac{11 - 4 + 3 + 0 + 18 - 1}{9}$

Step 2.4

Subtract $4$ from $11$ .

$\overline{x} = \frac{7 + 3 + 0 + 18 - 1}{9}$

Step 2.5

Add $7$ and $3$ .

$\overline{x} = \frac{10 + 0 + 18 - 1}{9}$

Step 2.6

Add $10$ and $0$ .

$\overline{x} = \frac{10 + 18 - 1}{9}$

Step 2.7

Add $10$ and $18$ .

$\overline{x} = \frac{28 - 1}{9}$

Step 2.8

Subtract $1$ from $28$ .

$\overline{x} = \frac{27}{9}$

Step 3

Divide $27$ by $9$ .

$\overline{x} = 3$

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.

$s = \frac{{(7 - 3)}^{2} + {(2 - 3)}^{2} + {(6 - 3)}^{2} + {(- 4 - 3)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $3$ from $7$ .

$s = \frac{4^{2} + {(2 - 3)}^{2} + {(6 - 3)}^{2} + {(- 4 - 3)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.2

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

$s = \frac{16 + {(2 - 3)}^{2} + {(6 - 3)}^{2} + {(- 4 - 3)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.3

Subtract $3$ from $2$ .

$s = \frac{16 + {(- 1)}^{2} + {(6 - 3)}^{2} + {(- 4 - 3)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.4

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

$s = \frac{16 + 1 + {(6 - 3)}^{2} + {(- 4 - 3)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.5

Subtract $3$ from $6$ .

$s = \frac{16 + 1 + 3^{2} + {(- 4 - 3)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.6

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

$s = \frac{16 + 1 + 9 + {(- 4 - 3)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.7

Subtract $3$ from $- 4$ .

$s = \frac{16 + 1 + 9 + {(- 7)}^{2} + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.8

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

$s = \frac{16 + 1 + 9 + 49 + {(- 4 - 3)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.9

Subtract $3$ from $- 4$ .

$s = \frac{16 + 1 + 9 + 49 + {(- 7)}^{2} + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.10

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

$s = \frac{16 + 1 + 9 + 49 + 49 + {(3 - 3)}^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.11

Subtract $3$ from $3$ .

$s = \frac{16 + 1 + 9 + 49 + 49 + 0^{2} + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.12

Raising $0$ to any positive power yields $0$ .

$s = \frac{16 + 1 + 9 + 49 + 49 + 0 + {(0 - 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.13

Subtract $3$ from $0$ .

$s = \frac{16 + 1 + 9 + 49 + 49 + 0 + {(- 3)}^{2} + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.14

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

$s = \frac{16 + 1 + 9 + 49 + 49 + 0 + 9 + {(18 - 3)}^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.15

Subtract $3$ from $18$ .

$s = \frac{16 + 1 + 9 + 49 + 49 + 0 + 9 + 15^{2} + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.16

Raise $15$ to the power of $2$ .

$s = \frac{16 + 1 + 9 + 49 + 49 + 0 + 9 + 225 + {(- 1 - 3)}^{2}}{9 - 1}$

Step 6.1.17

Subtract $3$ from $- 1$ .

$s = \frac{16 + 1 + 9 + 49 + 49 + 0 + 9 + 225 + {(- 4)}^{2}}{9 - 1}$

Step 6.1.18

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

$s = \frac{16 + 1 + 9 + 49 + 49 + 0 + 9 + 225 + 16}{9 - 1}$

Step 6.1.19

Add $16$ and $1$ .

$s = \frac{17 + 9 + 49 + 49 + 0 + 9 + 225 + 16}{9 - 1}$

Step 6.1.20

Add $17$ and $9$ .

$s = \frac{26 + 49 + 49 + 0 + 9 + 225 + 16}{9 - 1}$

Step 6.1.21

Add $26$ and $49$ .

$s = \frac{75 + 49 + 0 + 9 + 225 + 16}{9 - 1}$

Step 6.1.22

Add $75$ and $49$ .

$s = \frac{124 + 0 + 9 + 225 + 16}{9 - 1}$

Step 6.1.23

Add $124$ and $0$ .

$s = \frac{124 + 9 + 225 + 16}{9 - 1}$

Step 6.1.24

Add $124$ and $9$ .

$s = \frac{133 + 225 + 16}{9 - 1}$

Step 6.1.25

Add $133$ and $225$ .

$s = \frac{358 + 16}{9 - 1}$

Step 6.1.26

Add $358$ and $16$ .

$s = \frac{374}{9 - 1}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Subtract $1$ from $9$ .

$s = \frac{374}{8}$

Step 6.2.2

Cancel the common factor of $374$ and $8$ .

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Step 6.2.2.1

Factor $2$ out of $374$ .

$s = \frac{2 (187)}{8}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $2$ out of $8$ .

$s = \frac{2 \cdot 187}{2 \cdot 4}$

Step 6.2.2.2.2

Cancel the common factor.

$s = \frac{2 \cdot 187}{2 \cdot 4}$

Step 6.2.2.2.3

Rewrite the expression.

$s = \frac{187}{4}$

Step 7

Approximate the result.

$s^{2} \approx 46.75$