Statistics Examples

${2, 4, 6, 8, 10, 12}$

Step 1

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

$\overline{x} = \frac{2 + 4 + 6 + 8 + 10 + 12}{6}$

Step 2

Cancel the common factor of $2 + 4 + 6 + 8 + 10 + 12$ and $6$ .

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Step 2.1

Factor $2$ out of $2$ .

$\overline{x} = \frac{2 (1) + 4 + 6 + 8 + 10 + 12}{6}$

Step 2.2

Factor $2$ out of $4$ .

$\overline{x} = \frac{2 \cdot 1 + 2 \cdot 2 + 6 + 8 + 10 + 12}{6}$

Step 2.3

Factor $2$ out of $2 \cdot 1 + 2 \cdot 2$ .

$\overline{x} = \frac{2 \cdot (1 + 2) + 6 + 8 + 10 + 12}{6}$

Step 2.4

Factor $2$ out of $6$ .

$\overline{x} = \frac{2 \cdot (1 + 2) + 2 (3) + 8 + 10 + 12}{6}$

Step 2.5

Factor $2$ out of $2 \cdot (1 + 2) + 2 (3)$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3) + 8 + 10 + 12}{6}$

Step 2.6

Factor $2$ out of $8$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3) + 2 (4) + 10 + 12}{6}$

Step 2.7

Factor $2$ out of $2 \cdot (1 + 2 + 3) + 2 (4)$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3 + 4) + 10 + 12}{6}$

Step 2.8

Factor $2$ out of $10$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3 + 4) + 2 (5) + 12}{6}$

Step 2.9

Factor $2$ out of $2 \cdot (1 + 2 + 3 + 4) + 2 (5)$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3 + 4 + 5) + 12}{6}$

Step 2.10

Factor $2$ out of $12$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3 + 4 + 5) + 2 (6)}{6}$

Step 2.11

Factor $2$ out of $2 \cdot (1 + 2 + 3 + 4 + 5) + 2 (6)$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3 + 4 + 5 + 6)}{6}$

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $2$ out of $6$ .

$\overline{x} = \frac{2 \cdot (1 + 2 + 3 + 4 + 5 + 6)}{2 (3)}$

Step 2.12.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot (1 + 2 + 3 + 4 + 5 + 6)}{2 \cdot 3}$

Step 2.12.3

Rewrite the expression.

$\overline{x} = \frac{1 + 2 + 3 + 4 + 5 + 6}{3}$

Step 3

Simplify the numerator.

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Step 3.1

Add $1$ and $2$ .

$\overline{x} = \frac{3 + 3 + 4 + 5 + 6}{3}$

Step 3.2

Add $3$ and $3$ .

$\overline{x} = \frac{6 + 4 + 5 + 6}{3}$

Step 3.3

Add $6$ and $4$ .

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

Step 3.4

Add $10$ and $5$ .

$\overline{x} = \frac{15 + 6}{3}$

Step 3.5

Add $15$ and $6$ .

$\overline{x} = \frac{21}{3}$

Step 4

Divide $21$ by $3$ .

$\overline{x} = 7$

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.

$s = \frac{{(2 - 7)}^{2} + {(4 - 7)}^{2} + {(6 - 7)}^{2} + {(8 - 7)}^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $7$ from $2$ .

$s = \frac{{(- 5)}^{2} + {(4 - 7)}^{2} + {(6 - 7)}^{2} + {(8 - 7)}^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.2

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

$s = \frac{25 + {(4 - 7)}^{2} + {(6 - 7)}^{2} + {(8 - 7)}^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.3

Subtract $7$ from $4$ .

$s = \frac{25 + {(- 3)}^{2} + {(6 - 7)}^{2} + {(8 - 7)}^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.4

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

$s = \frac{25 + 9 + {(6 - 7)}^{2} + {(8 - 7)}^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.5

Subtract $7$ from $6$ .

$s = \frac{25 + 9 + {(- 1)}^{2} + {(8 - 7)}^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.6

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

$s = \frac{25 + 9 + 1 + {(8 - 7)}^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.7

Subtract $7$ from $8$ .

$s = \frac{25 + 9 + 1 + 1^{2} + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.8

One to any power is one.

$s = \frac{25 + 9 + 1 + 1 + {(10 - 7)}^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.9

Subtract $7$ from $10$ .

$s = \frac{25 + 9 + 1 + 1 + 3^{2} + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.10

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

$s = \frac{25 + 9 + 1 + 1 + 9 + {(12 - 7)}^{2}}{6 - 1}$

Step 7.1.11

Subtract $7$ from $12$ .

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

Step 7.1.12

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

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

Step 7.1.13

Add $25$ and $9$ .

$s = \frac{34 + 1 + 1 + 9 + 25}{6 - 1}$

Step 7.1.14

Add $34$ and $1$ .

$s = \frac{35 + 1 + 9 + 25}{6 - 1}$

Step 7.1.15

Add $35$ and $1$ .

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

Step 7.1.16

Add $36$ and $9$ .

$s = \frac{45 + 25}{6 - 1}$

Step 7.1.17

Add $45$ and $25$ .

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

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.

$s^{2} \approx 14$