Basic Math Examples

$72$ , $36$ , $40$ , $20$ , $24$ , $12$ , $16$

Step 1

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

$\overline{x} = \frac{72 + 36 + 40 + 20 + 24 + 12 + 16}{7}$

Step 2

Simplify the numerator.

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Step 2.1

Add $72$ and $36$ .

$\overline{x} = \frac{108 + 40 + 20 + 24 + 12 + 16}{7}$

Step 2.2

Add $108$ and $40$ .

$\overline{x} = \frac{148 + 20 + 24 + 12 + 16}{7}$

Step 2.3

Add $148$ and $20$ .

$\overline{x} = \frac{168 + 24 + 12 + 16}{7}$

Step 2.4

Add $168$ and $24$ .

$\overline{x} = \frac{192 + 12 + 16}{7}$

Step 2.5

Add $192$ and $12$ .

$\overline{x} = \frac{204 + 16}{7}$

Step 2.6

Add $204$ and $16$ .

$\overline{x} = \frac{220}{7}$

Step 3

Divide.

$\overline{x} = 31.42857142$

Step 4

The mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$\overline{x} = 31.4$

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.

$s = \frac{{(72 - 31.4)}^{2} + {(36 - 31.4)}^{2} + {(40 - 31.4)}^{2} + {(20 - 31.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $31.4$ from $72$ .

$s = \frac{{40.6}^{2} + {(36 - 31.4)}^{2} + {(40 - 31.4)}^{2} + {(20 - 31.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.2

Raise $40.6$ to the power of $2$ .

$s = \frac{1648.36 + {(36 - 31.4)}^{2} + {(40 - 31.4)}^{2} + {(20 - 31.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.3

Subtract $31.4$ from $36$ .

$s = \frac{1648.36 + {4.6}^{2} + {(40 - 31.4)}^{2} + {(20 - 31.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.4

Raise $4.6$ to the power of $2$ .

$s = \frac{1648.36 + 21.16 + {(40 - 31.4)}^{2} + {(20 - 31.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.5

Subtract $31.4$ from $40$ .

$s = \frac{1648.36 + 21.16 + {8.6}^{2} + {(20 - 31.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.6

Raise $8.6$ to the power of $2$ .

$s = \frac{1648.36 + 21.16 + 73.96 + {(20 - 31.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.7

Subtract $31.4$ from $20$ .

$s = \frac{1648.36 + 21.16 + 73.96 + {(- 11.4)}^{2} + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.8

Raise $- 11.4$ to the power of $2$ .

$s = \frac{1648.36 + 21.16 + 73.96 + 129.96 + {(24 - 31.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.9

Subtract $31.4$ from $24$ .

$s = \frac{1648.36 + 21.16 + 73.96 + 129.96 + {(- 7.4)}^{2} + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.10

Raise $- 7.4$ to the power of $2$ .

$s = \frac{1648.36 + 21.16 + 73.96 + 129.96 + 54.76 + {(12 - 31.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.11

Subtract $31.4$ from $12$ .

$s = \frac{1648.36 + 21.16 + 73.96 + 129.96 + 54.76 + {(- 19.4)}^{2} + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.12

Raise $- 19.4$ to the power of $2$ .

$s = \frac{1648.36 + 21.16 + 73.96 + 129.96 + 54.76 + 376.36 + {(16 - 31.4)}^{2}}{7 - 1}$

Step 7.1.13

Subtract $31.4$ from $16$ .

$s = \frac{1648.36 + 21.16 + 73.96 + 129.96 + 54.76 + 376.36 + {(- 15.4)}^{2}}{7 - 1}$

Step 7.1.14

Raise $- 15.4$ to the power of $2$ .

$s = \frac{1648.36 + 21.16 + 73.96 + 129.96 + 54.76 + 376.36 + 237.16}{7 - 1}$

Step 7.1.15

Add $1648.36$ and $21.16$ .

$s = \frac{1669.52 + 73.96 + 129.96 + 54.76 + 376.36 + 237.16}{7 - 1}$

Step 7.1.16

Add $1669.52$ and $73.96$ .

$s = \frac{1743.48 + 129.96 + 54.76 + 376.36 + 237.16}{7 - 1}$

Step 7.1.17

Add $1743.48$ and $129.96$ .

$s = \frac{1873.44 + 54.76 + 376.36 + 237.16}{7 - 1}$

Step 7.1.18

Add $1873.44$ and $54.76$ .

$s = \frac{1928.2 + 376.36 + 237.16}{7 - 1}$

Step 7.1.19

Add $1928.2$ and $376.36$ .

$s = \frac{2304.56 + 237.16}{7 - 1}$

Step 7.1.20

Add $2304.56$ and $237.16$ .

$s = \frac{2541.72}{7 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $7$ .

$s = \frac{2541.72}{6}$

Step 7.2.2

Divide $2541.72$ by $6$ .

$s = 423.62$

Step 8

Approximate the result.

$s^{2} \approx 423.62$