Algebra Examples

4

$4$ ,

6

$6$ ,

8

$8$ ,

12

$12$ ,

15

$15$ ,

17

$17$ ,

10

$10$ ,

24

$24$

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.

¯ x = \frac{4 + 6 + 8 + 12 + 15 + 17 + 10 + 24}{8}

$\overline{x} = \frac{4 + 6 + 8 + 12 + 15 + 17 + 10 + 24}{8}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Add

4

$4$ and

6

$6$ .

¯ x = \frac{10 + 8 + 12 + 15 + 17 + 10 + 24}{8}

$\overline{x} = \frac{10 + 8 + 12 + 15 + 17 + 10 + 24}{8}$

Step 1.2.2

Add

10

$10$ and

8

$8$ .

¯ x = \frac{18 + 12 + 15 + 17 + 10 + 24}{8}

$\overline{x} = \frac{18 + 12 + 15 + 17 + 10 + 24}{8}$

Step 1.2.3

Add

18

$18$ and

12

$12$ .

¯ x = \frac{30 + 15 + 17 + 10 + 24}{8}

$\overline{x} = \frac{30 + 15 + 17 + 10 + 24}{8}$

Step 1.2.4

Add

30

$30$ and

15

$15$ .

¯ x = \frac{45 + 17 + 10 + 24}{8}

$\overline{x} = \frac{45 + 17 + 10 + 24}{8}$

Step 1.2.5

Add

45

$45$ and

17

$17$ .

¯ x = \frac{62 + 10 + 24}{8}

$\overline{x} = \frac{62 + 10 + 24}{8}$

Step 1.2.6

Add

62

$62$ and

10

$10$ .

¯ x = \frac{72 + 24}{8}

$\overline{x} = \frac{72 + 24}{8}$

Step 1.2.7

Add

72

$72$ and

24

$24$ .

¯ x = \frac{96}{8}

$\overline{x} = \frac{96}{8}$

¯ x = \frac{96}{8}

$\overline{x} = \frac{96}{8}$

Step 1.3

Divide

96

$96$ by

8

$8$ .

¯ x = 12

$\overline{x} = 12$

¯ x = 12

$\overline{x} = 12$

Step 2

Simplify each value in the list.

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Step 2.1

Convert

4

$4$ to a decimal value.

4

$4$

Step 2.2

Convert

6

$6$ to a decimal value.

6

$6$

Step 2.3

Convert

8

$8$ to a decimal value.

8

$8$

Step 2.4

Convert

12

$12$ to a decimal value.

12

$12$

Step 2.5

Convert

15

$15$ to a decimal value.

15

$15$

Step 2.6

Convert

17

$17$ to a decimal value.

17

$17$

Step 2.7

Convert

10

$10$ to a decimal value.

10

$10$

Step 2.8

Convert

24

$24$ to a decimal value.

24

$24$

Step 2.9

The simplified values are

4, 6, 8, 12, 15, 17, 10, 24

$4, 6, 8, 12, 15, 17, 10, 24$ .

4, 6, 8, 12, 15, 17, 10, 24

$4, 6, 8, 12, 15, 17, 10, 24$

4, 6, 8, 12, 15, 17, 10, 24

$4, 6, 8, 12, 15, 17, 10, 24$

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 = n \sum i = 1 \sqrt{\frac{{(x_{i} - x_{a v g})}_{i}^{2}}{n - 1}}

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

Step 4

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

s = \sqrt{\frac{{(4 - 12)}^{2} + {(6 - 12)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}

$s = \sqrt{\frac{{(4 - 12)}^{2} + {(6 - 12)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5

Simplify the result.

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Step 5.1

Subtract

12

$12$ from

4

$4$ .

s = \sqrt{\frac{{(- 8)}^{2} + {(6 - 12)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}

$s = \sqrt{\frac{{(- 8)}^{2} + {(6 - 12)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.2

Raise

- 8

$- 8$ to the power of

2

$2$ .

s = \sqrt{\frac{64 + {(6 - 12)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}

$s = \sqrt{\frac{64 + {(6 - 12)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.3

Subtract

12

$12$ from

6

$6$ .

s = \sqrt{\frac{64 + {(- 6)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}

$s = \sqrt{\frac{64 + {(- 6)}^{2} + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.4

Raise

- 6

$- 6$ to the power of

2

$2$ .

s = \sqrt{\frac{64 + 36 + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}

$s = \sqrt{\frac{64 + 36 + {(8 - 12)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.5

Subtract

12

$12$ from

8

$8$ .

s = \sqrt{\frac{64 + 36 + {(- 4)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}

$s = \sqrt{\frac{64 + 36 + {(- 4)}^{2} + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.6

Raise

- 4

$- 4$ to the power of

2

$2$ .

s = \sqrt{\frac{64 + 36 + 16 + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}

$s = \sqrt{\frac{64 + 36 + 16 + {(12 - 12)}^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.7

Subtract

$12$ from

$12$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0^{2} + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.8

Raising

$0$ to any positive power yields

$0$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + {(15 - 12)}^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.9

Subtract

$12$ from

$15$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 3^{2} + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.10

Raise

$3$ to the power of

$2$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 9 + {(17 - 12)}^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.11

Subtract

$12$ from

$17$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 9 + 5^{2} + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.12

Raise

$5$ to the power of

$2$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 9 + 25 + {(10 - 12)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.13

Subtract

$12$ from

$10$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 9 + 25 + {(- 2)}^{2} + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.14

Raise

$- 2$ to the power of

$2$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 9 + 25 + 4 + {(24 - 12)}^{2}}{8 - 1}}$

Step 5.15

Subtract

$12$ from

$24$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 9 + 25 + 4 + 12^{2}}{8 - 1}}$

Step 5.16

Raise

$12$ to the power of

$2$ .

$s = \sqrt{\frac{64 + 36 + 16 + 0 + 9 + 25 + 4 + 144}{8 - 1}}$

Step 5.17

Add

$64$ and

$36$ .

$s = \sqrt{\frac{100 + 16 + 0 + 9 + 25 + 4 + 144}{8 - 1}}$

Step 5.18

Add

$100$ and

$16$ .

$s = \sqrt{\frac{116 + 0 + 9 + 25 + 4 + 144}{8 - 1}}$

Step 5.19

Add

$116$ and

$0$ .

$s = \sqrt{\frac{116 + 9 + 25 + 4 + 144}{8 - 1}}$

Step 5.20

Add

$116$ and

$9$ .

$s = \sqrt{\frac{125 + 25 + 4 + 144}{8 - 1}}$

Step 5.21

Add

$125$ and

$25$ .

$s = \sqrt{\frac{150 + 4 + 144}{8 - 1}}$

Step 5.22

Add

$150$ and

$4$ .

$s = \sqrt{\frac{154 + 144}{8 - 1}}$

Step 5.23

Add

$154$ and

$144$ .

$s = \sqrt{\frac{298}{8 - 1}}$

Step 5.24

Subtract

$1$ from

$8$ .

$s = \sqrt{\frac{298}{7}}$

Step 5.25

Rewrite

$\sqrt{\frac{298}{7}}$ as

$\frac{\sqrt{298}}{\sqrt{7}}$ .

$s = \frac{\sqrt{298}}{\sqrt{7}}$

Step 5.26

Multiply

$\frac{\sqrt{298}}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$s = \frac{\sqrt{298}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 5.27

Combine and simplify the denominator.

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Step 5.27.1

Multiply

$\frac{\sqrt{298}}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$s = \frac{\sqrt{298} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 5.27.2

Raise

$\sqrt{7}$ to the power of

$1$ .

$s = \frac{\sqrt{298} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 5.27.3

Raise

$\sqrt{7}$ to the power of

$1$ .

$s = \frac{\sqrt{298} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 5.27.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s = \frac{\sqrt{298} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 5.27.5

Add

$1$ and

$1$ .

$s = \frac{\sqrt{298} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 5.27.6

Rewrite

${\sqrt{7}}^{2}$ as

$7$ .

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Step 5.27.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{7}$ as

$7^{\frac{1}{2}}$ .

$s = \frac{\sqrt{298} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 5.27.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{\sqrt{298} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 5.27.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$s = \frac{\sqrt{298} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.27.6.4

Cancel the common factor of

$2$ .

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Step 5.27.6.4.1

Cancel the common factor.

$s = \frac{\sqrt{298} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.27.6.4.2

Rewrite the expression.

$s = \frac{\sqrt{298} \sqrt{7}}{7}$

Step 5.27.6.5

Evaluate the exponent.

$s = \frac{\sqrt{298} \sqrt{7}}{7}$

Step 5.28

Simplify the numerator.

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Step 5.28.1

Combine using the product rule for radicals.

$s = \frac{\sqrt{298 \cdot 7}}{7}$

Step 5.28.2

Multiply

$298$ by

$7$ .

$s = \frac{\sqrt{2086}}{7}$

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.

$6.5$