Basic Math Examples

$14$ , $12$ , $16$ , $20$ , $10$ , $18$

Step 1

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

$\overline{x} = \frac{14 + 12 + 16 + 20 + 10 + 18}{6}$

Step 2

Cancel the common factor of $14 + 12 + 16 + 20 + 10 + 18$ and $6$ .

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Step 2.1

Factor $2$ out of $14$ .

$\overline{x} = \frac{2 (7) + 12 + 16 + 20 + 10 + 18}{6}$

Step 2.2

Factor $2$ out of $12$ .

$\overline{x} = \frac{2 \cdot 7 + 2 \cdot 6 + 16 + 20 + 10 + 18}{6}$

Step 2.3

Factor $2$ out of $2 \cdot 7 + 2 \cdot 6$ .

$\overline{x} = \frac{2 \cdot (7 + 6) + 16 + 20 + 10 + 18}{6}$

Step 2.4

Factor $2$ out of $16$ .

$\overline{x} = \frac{2 \cdot (7 + 6) + 2 (8) + 20 + 10 + 18}{6}$

Step 2.5

Factor $2$ out of $2 \cdot (7 + 6) + 2 (8)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8) + 20 + 10 + 18}{6}$

Step 2.6

Factor $2$ out of $20$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8) + 2 (10) + 10 + 18}{6}$

Step 2.7

Factor $2$ out of $2 \cdot (7 + 6 + 8) + 2 (10)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10) + 10 + 18}{6}$

Step 2.8

Factor $2$ out of $10$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10) + 2 (5) + 18}{6}$

Step 2.9

Factor $2$ out of $2 \cdot (7 + 6 + 8 + 10) + 2 (5)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5) + 18}{6}$

Step 2.10

Factor $2$ out of $18$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5) + 2 (9)}{6}$

Step 2.11

Factor $2$ out of $2 \cdot (7 + 6 + 8 + 10 + 5) + 2 (9)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5 + 9)}{6}$

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $2$ out of $6$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5 + 9)}{2 (3)}$

Step 2.12.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5 + 9)}{2 \cdot 3}$

Step 2.12.3

Rewrite the expression.

$\overline{x} = \frac{7 + 6 + 8 + 10 + 5 + 9}{3}$

Step 3

Simplify the numerator.

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Step 3.1

Add $7$ and $6$ .

$\overline{x} = \frac{13 + 8 + 10 + 5 + 9}{3}$

Step 3.2

Add $13$ and $8$ .

$\overline{x} = \frac{21 + 10 + 5 + 9}{3}$

Step 3.3

Add $21$ and $10$ .

$\overline{x} = \frac{31 + 5 + 9}{3}$

Step 3.4

Add $31$ and $5$ .

$\overline{x} = \frac{36 + 9}{3}$

Step 3.5

Add $36$ and $9$ .

$\overline{x} = \frac{45}{3}$

Step 4

Divide $45$ by $3$ .

$\overline{x} = 15$

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.

$s = \frac{{(14 - 15)}^{2} + {(12 - 15)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $15$ from $14$ .

$s = \frac{{(- 1)}^{2} + {(12 - 15)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.2

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

$s = \frac{1 + {(12 - 15)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.3

Subtract $15$ from $12$ .

$s = \frac{1 + {(- 3)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.4

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

$s = \frac{1 + 9 + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.5

Subtract $15$ from $16$ .

$s = \frac{1 + 9 + 1^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.6

One to any power is one.

$s = \frac{1 + 9 + 1 + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.7

Subtract $15$ from $20$ .

$s = \frac{1 + 9 + 1 + 5^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.8

Raise $5$ to the power of $2$ .

$s = \frac{1 + 9 + 1 + 25 + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.9

Subtract $15$ from $10$ .

$s = \frac{1 + 9 + 1 + 25 + {(- 5)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.10

Raise $- 5$ to the power of $2$ .

$s = \frac{1 + 9 + 1 + 25 + 25 + {(18 - 15)}^{2}}{6 - 1}$

Step 7.1.11

Subtract $15$ from $18$ .

$s = \frac{1 + 9 + 1 + 25 + 25 + 3^{2}}{6 - 1}$

Step 7.1.12

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

$s = \frac{1 + 9 + 1 + 25 + 25 + 9}{6 - 1}$

Step 7.1.13

Add $1$ and $9$ .

$s = \frac{10 + 1 + 25 + 25 + 9}{6 - 1}$

Step 7.1.14

Add $10$ and $1$ .

$s = \frac{11 + 25 + 25 + 9}{6 - 1}$

Step 7.1.15

Add $11$ and $25$ .

$s = \frac{36 + 25 + 9}{6 - 1}$

Step 7.1.16

Add $36$ and $25$ .

$s = \frac{61 + 9}{6 - 1}$

Step 7.1.17

Add $61$ and $9$ .

$s = \frac{70}{6 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $6$ .

$s = \frac{70}{5}$

Step 7.2.2

Divide $70$ by $5$ .

$s = 14$

Step 8

Approximate the result.

$s^{2} \approx 14$