Basic Math Examples

$73$ , $79$ , $79$ , $79$ , $80$ , $81$ , $83$ , $85$ , $86$ , $87$ , $90$

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.

$\overline{x} = \frac{73 + 79 + 79 + 79 + 80 + 81 + 83 + 85 + 86 + 87 + 90}{11}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Add $73$ and $79$ .

$\overline{x} = \frac{152 + 79 + 79 + 80 + 81 + 83 + 85 + 86 + 87 + 90}{11}$

Step 1.2.2

Add $152$ and $79$ .

$\overline{x} = \frac{231 + 79 + 80 + 81 + 83 + 85 + 86 + 87 + 90}{11}$

Step 1.2.3

Add $231$ and $79$ .

$\overline{x} = \frac{310 + 80 + 81 + 83 + 85 + 86 + 87 + 90}{11}$

Step 1.2.4

Add $310$ and $80$ .

$\overline{x} = \frac{390 + 81 + 83 + 85 + 86 + 87 + 90}{11}$

Step 1.2.5

Add $390$ and $81$ .

$\overline{x} = \frac{471 + 83 + 85 + 86 + 87 + 90}{11}$

Step 1.2.6

Add $471$ and $83$ .

$\overline{x} = \frac{554 + 85 + 86 + 87 + 90}{11}$

Step 1.2.7

Add $554$ and $85$ .

$\overline{x} = \frac{639 + 86 + 87 + 90}{11}$

Step 1.2.8

Add $639$ and $86$ .

$\overline{x} = \frac{725 + 87 + 90}{11}$

Step 1.2.9

Add $725$ and $87$ .

$\overline{x} = \frac{812 + 90}{11}$

Step 1.2.10

Add $812$ and $90$ .

$\overline{x} = \frac{902}{11}$

Step 1.3

Divide $902$ by $11$ .

$\overline{x} = 82$

Step 2

Simplify each value in the list.

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Step 2.1

Convert $73$ to a decimal value.

$73$

Step 2.2

Convert $79$ to a decimal value.

$79$

Step 2.3

Convert $80$ to a decimal value.

$80$

Step 2.4

Convert $81$ to a decimal value.

$81$

Step 2.5

Convert $83$ to a decimal value.

$83$

Step 2.6

Convert $85$ to a decimal value.

$85$

Step 2.7

Convert $86$ to a decimal value.

$86$

Step 2.8

Convert $87$ to a decimal value.

$87$

Step 2.9

Convert $90$ to a decimal value.

$90$

Step 2.10

The simplified values are $73, 79, 79, 79, 80, 81, 83, 85, 86, 87, 90$ .

$73, 79, 79, 79, 80, 81, 83, 85, 86, 87, 90$

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.

$s = \sqrt{\frac{{(73 - 82)}^{2} + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5

Simplify the result.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Subtract $82$ from $73$ .

$s = \sqrt{\frac{{(- 9)}^{2} + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.2

Raise $- 9$ to the power of $2$ .

$s = \sqrt{\frac{81 + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.3

Subtract $82$ from $79$ .

$s = \sqrt{\frac{81 + {(- 3)}^{2} + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.4

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

$s = \sqrt{\frac{81 + 9 + {(79 - 82)}^{2} + {(79 - 82)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.5

Subtract $82$ from $79$ .

$s = \sqrt{\frac{81 + 9 + {(- 3)}^{2} + {(79 - 82)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.6

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

$s = \sqrt{\frac{81 + 9 + 9 + {(79 - 82)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.7

Subtract $82$ from $79$ .

$s = \sqrt{\frac{81 + 9 + 9 + {(- 3)}^{2} + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.8

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

$s = \sqrt{\frac{81 + 9 + 9 + 9 + {(80 - 82)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.9

Subtract $82$ from $80$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + {(- 2)}^{2} + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.10

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

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + {(81 - 82)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.11

Subtract $82$ from $81$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + {(- 1)}^{2} + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.12

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

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + {(83 - 82)}^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.13

Subtract $82$ from $83$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1^{2} + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.14

One to any power is one.

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + {(85 - 82)}^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.15

Subtract $82$ from $85$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 3^{2} + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.16

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

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 9 + {(86 - 82)}^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.17

Subtract $82$ from $86$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 9 + 4^{2} + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.18

Raise $4$ to the power of $2$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 9 + 16 + {(87 - 82)}^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.19

Subtract $82$ from $87$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 9 + 16 + 5^{2} + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.20

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

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 9 + 16 + 25 + {(90 - 82)}^{2}}{11 - 1}}$

Step 5.1.21

Subtract $82$ from $90$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 9 + 16 + 25 + 8^{2}}{11 - 1}}$

Step 5.1.22

Raise $8$ to the power of $2$ .

$s = \sqrt{\frac{81 + 9 + 9 + 9 + 4 + 1 + 1 + 9 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.23

Add $81$ and $9$ .

$s = \sqrt{\frac{90 + 9 + 9 + 4 + 1 + 1 + 9 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.24

Add $90$ and $9$ .

$s = \sqrt{\frac{99 + 9 + 4 + 1 + 1 + 9 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.25

Add $99$ and $9$ .

$s = \sqrt{\frac{108 + 4 + 1 + 1 + 9 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.26

Add $108$ and $4$ .

$s = \sqrt{\frac{112 + 1 + 1 + 9 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.27

Add $112$ and $1$ .

$s = \sqrt{\frac{113 + 1 + 9 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.28

Add $113$ and $1$ .

$s = \sqrt{\frac{114 + 9 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.29

Add $114$ and $9$ .

$s = \sqrt{\frac{123 + 16 + 25 + 64}{11 - 1}}$

Step 5.1.30

Add $123$ and $16$ .

$s = \sqrt{\frac{139 + 25 + 64}{11 - 1}}$

Step 5.1.31

Add $139$ and $25$ .

$s = \sqrt{\frac{164 + 64}{11 - 1}}$

Step 5.1.32

Add $164$ and $64$ .

$s = \sqrt{\frac{228}{11 - 1}}$

Step 5.1.33

Subtract $1$ from $11$ .

$s = \sqrt{\frac{228}{10}}$

Step 5.2

Cancel the common factor of $228$ and $10$ .

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Step 5.2.1

Factor $2$ out of $228$ .

$s = \sqrt{\frac{2 (114)}{10}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $10$ .

$s = \sqrt{\frac{2 \cdot 114}{2 \cdot 5}}$

Step 5.2.2.2

Cancel the common factor.

$s = \sqrt{\frac{2 \cdot 114}{2 \cdot 5}}$

Step 5.2.2.3

Rewrite the expression.

$s = \sqrt{\frac{114}{5}}$

Step 5.3

Rewrite $\sqrt{\frac{114}{5}}$ as $\frac{\sqrt{114}}{\sqrt{5}}$ .

$s = \frac{\sqrt{114}}{\sqrt{5}}$

Step 5.4

Multiply $\frac{\sqrt{114}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$s = \frac{\sqrt{114}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.5

Combine and simplify the denominator.

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Step 5.5.1

Multiply $\frac{\sqrt{114}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$s = \frac{\sqrt{114} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$s = \frac{\sqrt{114} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$s = \frac{\sqrt{114} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s = \frac{\sqrt{114} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 5.5.5

Add $1$ and $1$ .

$s = \frac{\sqrt{114} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 5.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$s = \frac{\sqrt{114} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 5.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{\sqrt{114} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 5.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$s = \frac{\sqrt{114} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.5.6.4

Cancel the common factor of $2$ .

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Step 5.5.6.4.1

Cancel the common factor.

$s = \frac{\sqrt{114} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.5.6.4.2

Rewrite the expression.

$s = \frac{\sqrt{114} \sqrt{5}}{5}$

Step 5.5.6.5

Evaluate the exponent.

$s = \frac{\sqrt{114} \sqrt{5}}{5}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Combine using the product rule for radicals.

$s = \frac{\sqrt{114 \cdot 5}}{5}$

Step 5.6.2

Multiply $114$ by $5$ .

$s = \frac{\sqrt{570}}{5}$

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.

$4.8$