Basic Math Examples

15

$15$ ,

15.9

$15.9$ ,

15.3

$15.3$ ,

15.3

$15.3$ ,

15.5

$15.5$ ,

15.9

$15.9$ ,

15.9

$15.9$ ,

15

$15$

Step 1

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

¯ x = \frac{15 + 15.9 + 15.3 + 15.3 + 15.5 + 15.9 + 15.9 + 15}{8}

$\overline{x} = \frac{15 + 15.9 + 15.3 + 15.3 + 15.5 + 15.9 + 15.9 + 15}{8}$

Step 2

Simplify the numerator.

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Step 2.1

Add

15

$15$ and

15.9

$15.9$ .

¯ x = \frac{30.9 + 15.3 + 15.3 + 15.5 + 15.9 + 15.9 + 15}{8}

$\overline{x} = \frac{30.9 + 15.3 + 15.3 + 15.5 + 15.9 + 15.9 + 15}{8}$

Step 2.2

Add

30.9

$30.9$ and

15.3

$15.3$ .

¯ x = \frac{46.2 + 15.3 + 15.5 + 15.9 + 15.9 + 15}{8}

$\overline{x} = \frac{46.2 + 15.3 + 15.5 + 15.9 + 15.9 + 15}{8}$

Step 2.3

Add

46.2

$46.2$ and

15.3

$15.3$ .

¯ x = \frac{61.5 + 15.5 + 15.9 + 15.9 + 15}{8}

$\overline{x} = \frac{61.5 + 15.5 + 15.9 + 15.9 + 15}{8}$

Step 2.4

Add

61.5

$61.5$ and

15.5

$15.5$ .

¯ x = \frac{77 + 15.9 + 15.9 + 15}{8}

$\overline{x} = \frac{77 + 15.9 + 15.9 + 15}{8}$

Step 2.5

Add

$77$ and

$15.9$ .

$\overline{x} = \frac{92.9 + 15.9 + 15}{8}$

Step 2.6

Add

$92.9$ and

$15.9$ .

$\overline{x} = \frac{108.8 + 15}{8}$

Step 2.7

Add

$108.8$ and

$15$ .

$\overline{x} = \frac{123.8}{8}$

Step 3

Divide

$123.8$ by

$8$ .

$\overline{x} = 15.475$

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.

$\overline{x} = 15.48$

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.

$s = \frac{{(15 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract

$15.48$ from

$15$ .

$s = \frac{{(- 0.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.2

Raise

$- 0.48$ to the power of

$2$ .

$s = \frac{0.2304 + {(15.9 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.3

Subtract

$15.48$ from

$15.9$ .

$s = \frac{0.2304 + {0.42}^{2} + {(15.3 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.4

Raise

$0.42$ to the power of

$2$ .

$s = \frac{0.2304 + 0.1764 + {(15.3 - 15.48)}^{2} + {(15.3 - 15.48)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.5

Subtract

$15.48$ from

$15.3$ .

$s = \frac{0.2304 + 0.1764 + {(- 0.18)}^{2} + {(15.3 - 15.48)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.6

Raise

$- 0.18$ to the power of

$2$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + {(15.3 - 15.48)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.7

Subtract

$15.48$ from

$15.3$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + {(- 0.18)}^{2} + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.8

Raise

$- 0.18$ to the power of

$2$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + {(15.5 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.9

Subtract

$15.48$ from

$15.5$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + {0.02}^{2} + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.10

Raise

$0.02$ to the power of

$2$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + 0.0004 + {(15.9 - 15.48)}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.11

Subtract

$15.48$ from

$15.9$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + 0.0004 + {0.42}^{2} + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.12

Raise

$0.42$ to the power of

$2$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + 0.0004 + 0.1764 + {(15.9 - 15.48)}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.13

Subtract

$15.48$ from

$15.9$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + 0.0004 + 0.1764 + {0.42}^{2} + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.14

Raise

$0.42$ to the power of

$2$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + 0.0004 + 0.1764 + 0.1764 + {(15 - 15.48)}^{2}}{8 - 1}$

Step 7.1.15

Subtract

$15.48$ from

$15$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + 0.0004 + 0.1764 + 0.1764 + {(- 0.48)}^{2}}{8 - 1}$

Step 7.1.16

Raise

$- 0.48$ to the power of

$2$ .

$s = \frac{0.2304 + 0.1764 + 0.0324 + 0.0324 + 0.0004 + 0.1764 + 0.1764 + 0.2304}{8 - 1}$

Step 7.1.17

Add

$0.2304$ and

$0.1764$ .

$s = \frac{0.4068 + 0.0324 + 0.0324 + 0.0004 + 0.1764 + 0.1764 + 0.2304}{8 - 1}$

Step 7.1.18

Add

$0.4068$ and

$0.0324$ .

$s = \frac{0.4392 + 0.0324 + 0.0004 + 0.1764 + 0.1764 + 0.2304}{8 - 1}$

Step 7.1.19

Add

$0.4392$ and

$0.0324$ .

$s = \frac{0.4716 + 0.0004 + 0.1764 + 0.1764 + 0.2304}{8 - 1}$

Step 7.1.20

Add

$0.4716$ and

$0.0004$ .

$s = \frac{0.472 + 0.1764 + 0.1764 + 0.2304}{8 - 1}$

Step 7.1.21

Add

$0.472$ and

$0.1764$ .

$s = \frac{0.6484 + 0.1764 + 0.2304}{8 - 1}$

Step 7.1.22

Add

$0.6484$ and

$0.1764$ .

$s = \frac{0.8248 + 0.2304}{8 - 1}$

Step 7.1.23

Add

$0.8248$ and

$0.2304$ .

$s = \frac{1.0552}{8 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract

$1$ from

$8$ .

$s = \frac{1.0552}{7}$

Step 7.2.2

Divide

$1.0552$ by

$7$ .

$s = 0.15074285$

Step 8

Approximate the result.

$s^{2} \approx 0.1507$