Basic Math Examples

$72$ ,

$90$ ,

$85$ ,

$84$ ,

$91$ ,

$78$ ,

$94$ ,

$86$ ,

$82$ ,

$78$ ,

$86$ ,

$78$ ,

$95$ ,

$90$ ,

$88$

Step 1

Find the mean.

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Step 1.1

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

$\overline{x} = \frac{72 + 90 + 85 + 84 + 91 + 78 + 94 + 86 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Add

$72$ and

$90$ .

$\overline{x} = \frac{162 + 85 + 84 + 91 + 78 + 94 + 86 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.2

Add

$162$ and

$85$ .

$\overline{x} = \frac{247 + 84 + 91 + 78 + 94 + 86 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.3

Add

$247$ and

$84$ .

$\overline{x} = \frac{331 + 91 + 78 + 94 + 86 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.4

Add

$331$ and

$91$ .

$\overline{x} = \frac{422 + 78 + 94 + 86 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.5

Add

$422$ and

$78$ .

$\overline{x} = \frac{500 + 94 + 86 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.6

Add

$500$ and

$94$ .

$\overline{x} = \frac{594 + 86 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.7

Add

$594$ and

$86$ .

$\overline{x} = \frac{680 + 82 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.8

Add

$680$ and

$82$ .

$\overline{x} = \frac{762 + 78 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.9

Add

$762$ and

$78$ .

$\overline{x} = \frac{840 + 86 + 78 + 95 + 90 + 88}{15}$

Step 1.2.10

Add

$840$ and

$86$ .

$\overline{x} = \frac{926 + 78 + 95 + 90 + 88}{15}$

Step 1.2.11

Add

$926$ and

$78$ .

$\overline{x} = \frac{1004 + 95 + 90 + 88}{15}$

Step 1.2.12

Add

$1004$ and

$95$ .

$\overline{x} = \frac{1099 + 90 + 88}{15}$

Step 1.2.13

Add

$1099$ and

$90$ .

$\overline{x} = \frac{1189 + 88}{15}$

Step 1.2.14

Add

$1189$ and

$88$ .

$\overline{x} = \frac{1277}{15}$

Step 1.3

Divide.

$\overline{x} = 85.1\overline{3}$

Step 1.4

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.

$\overline{x} = 85.1$

Step 2

Simplify each value in the list.

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Step 2.1

Convert

$72$ to a decimal value.

$72$

Step 2.2

Convert

$90$ to a decimal value.

$90$

Step 2.3

Convert

$85$ to a decimal value.

$85$

Step 2.4

Convert

$84$ to a decimal value.

$84$

Step 2.5

Convert

$91$ to a decimal value.

$91$

Step 2.6

Convert

$78$ to a decimal value.

$78$

Step 2.7

Convert

$94$ to a decimal value.

$94$

Step 2.8

Convert

$86$ to a decimal value.

$86$

Step 2.9

Convert

$82$ to a decimal value.

$82$

Step 2.10

Convert

$78$ to a decimal value.

$78$

Step 2.11

Convert

$86$ to a decimal value.

$86$

Step 2.12

Convert

$78$ to a decimal value.

$78$

Step 2.13

Convert

$95$ to a decimal value.

$95$

Step 2.14

Convert

$90$ to a decimal value.

$90$

Step 2.15

Convert

$88$ to a decimal value.

$88$

Step 2.16

The simplified values are

$72, 90, 85, 84, 91, 78, 94, 86, 82, 78, 86, 78, 95, 90, 88$ .

$72, 90, 85, 84, 91, 78, 94, 86, 82, 78, 86, 78, 95, 90, 88$

Step 3

Set up the formula for sample standard deviation. The standard deviation of a set of values is a measure of the spread of its values.

$s = \sum_{i = 1}^{n} \sqrt{\frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}}$

Step 4

Set up the formula for standard deviation for this set of numbers.

$s = \sqrt{\frac{{(72 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(85 - 85.1)}^{2} + {(84 - 85.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5

Simplify the result.

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Step 5.1

Subtract

$85.1$ from

$72$ .

$s = \sqrt{\frac{{(- 13.1)}^{2} + {(90 - 85.1)}^{2} + {(85 - 85.1)}^{2} + {(84 - 85.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.2

Raise

$- 13.1$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + {(90 - 85.1)}^{2} + {(85 - 85.1)}^{2} + {(84 - 85.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.3

Subtract

$85.1$ from

$90$ .

$s = \sqrt{\frac{171.61 + {4.9}^{2} + {(85 - 85.1)}^{2} + {(84 - 85.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.4

Raise

$4.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + {(85 - 85.1)}^{2} + {(84 - 85.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.5

Subtract

$85.1$ from

$85$ .

$s = \sqrt{\frac{171.61 + 24.01 + {(- 0.1)}^{2} + {(84 - 85.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.6

Raise

$- 0.1$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + {(84 - 85.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.7

Subtract

$85.1$ from

$84$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + {(- 1.1)}^{2} + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.8

Raise

$- 1.1$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + {(91 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.9

Subtract

$85.1$ from

$91$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + {5.9}^{2} + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.10

Raise

$5.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + {(78 - 85.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.11

Subtract

$85.1$ from

$78$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + {(- 7.1)}^{2} + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.12

Raise

$- 7.1$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + {(94 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.13

Subtract

$85.1$ from

$94$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + {8.9}^{2} + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.14

Raise

$8.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + {(86 - 85.1)}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.15

Subtract

$85.1$ from

$86$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + {0.9}^{2} + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.16

Raise

$0.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + {(82 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.17

Subtract

$85.1$ from

$82$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + {(- 3.1)}^{2} + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.18

Raise

$- 3.1$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + {(78 - 85.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.19

Subtract

$85.1$ from

$78$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + {(- 7.1)}^{2} + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.20

Raise

$- 7.1$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + {(86 - 85.1)}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.21

Subtract

$85.1$ from

$86$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + {0.9}^{2} + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.22

Raise

$0.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + {(78 - 85.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.23

Subtract

$85.1$ from

$78$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + {(- 7.1)}^{2} + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.24

Raise

$- 7.1$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + {(95 - 85.1)}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.25

Subtract

$85.1$ from

$95$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + {9.9}^{2} + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.26

Raise

$9.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + {(90 - 85.1)}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.27

Subtract

$85.1$ from

$90$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + {4.9}^{2} + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.28

Raise

$4.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + {(88 - 85.1)}^{2}}{15 - 1}}$

Step 5.29

Subtract

$85.1$ from

$88$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + {2.9}^{2}}{15 - 1}}$

Step 5.30

Raise

$2.9$ to the power of

$2$ .

$s = \sqrt{\frac{171.61 + 24.01 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.31

Add

$171.61$ and

$24.01$ .

$s = \sqrt{\frac{195.62 + 0.01 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.32

Add

$195.62$ and

$0.01$ .

$s = \sqrt{\frac{195.63 + 1.21 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.33

Add

$195.63$ and

$1.21$ .

$s = \sqrt{\frac{196.84 + 34.81 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.34

Add

$196.84$ and

$34.81$ .

$s = \sqrt{\frac{231.65 + 50.41 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.35

Add

$231.65$ and

$50.41$ .

$s = \sqrt{\frac{282.06 + 79.21 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.36

Add

$282.06$ and

$79.21$ .

$s = \sqrt{\frac{361.27 + 0.81 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.37

Add

$361.27$ and

$0.81$ .

$s = \sqrt{\frac{362.08 + 9.61 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.38

Add

$362.08$ and

$9.61$ .

$s = \sqrt{\frac{371.69 + 50.41 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.39

Add

$371.69$ and

$50.41$ .

$s = \sqrt{\frac{422.1 + 0.81 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.40

Add

$422.1$ and

$0.81$ .

$s = \sqrt{\frac{422.91 + 50.41 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.41

Add

$422.91$ and

$50.41$ .

$s = \sqrt{\frac{473.32 + 98.01 + 24.01 + 8.41}{15 - 1}}$

Step 5.42

Add

$473.32$ and

$98.01$ .

$s = \sqrt{\frac{571.33 + 24.01 + 8.41}{15 - 1}}$

Step 5.43

Add

$571.33$ and

$24.01$ .

$s = \sqrt{\frac{595.34 + 8.41}{15 - 1}}$

Step 5.44

Add

$595.34$ and

$8.41$ .

$s = \sqrt{\frac{603.75}{15 - 1}}$

Step 5.45

Subtract

$1$ from

$15$ .

$s = \sqrt{\frac{603.75}{14}}$

Step 5.46

Divide

$603.75$ by

$14$ .

$s = \sqrt{43.125}$

Step 6

The standard deviation 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.

$6.6$