Algebra Examples

1

$1$ ,

$2$ ,

$3$ ,

$4$ ,

$5$ ,

$6$ ,

$7$ ,

$8$ ,

$9$

Step 1

Find the mean.

Tap for more steps...

Step 1.1

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

$\overline{x} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9}{9}$

Step 1.2

Simplify the numerator.

Tap for more steps...

Step 1.2.1

Add

$1$ and

$2$ .

$\overline{x} = \frac{3 + 3 + 4 + 5 + 6 + 7 + 8 + 9}{9}$

Step 1.2.2

Add

$3$ and

$3$ .

$\overline{x} = \frac{6 + 4 + 5 + 6 + 7 + 8 + 9}{9}$

Step 1.2.3

Add

$6$ and

$4$ .

$\overline{x} = \frac{10 + 5 + 6 + 7 + 8 + 9}{9}$

Step 1.2.4

Add

$10$ and

$5$ .

$\overline{x} = \frac{15 + 6 + 7 + 8 + 9}{9}$

Step 1.2.5

Add

$15$ and

$6$ .

$\overline{x} = \frac{21 + 7 + 8 + 9}{9}$

Step 1.2.6

Add

$21$ and

$7$ .

$\overline{x} = \frac{28 + 8 + 9}{9}$

Step 1.2.7

Add

$28$ and

$8$ .

$\overline{x} = \frac{36 + 9}{9}$

Step 1.2.8

Add

$36$ and

$9$ .

$\overline{x} = \frac{45}{9}$

Step 1.3

Divide

$45$ by

$9$ .

$\overline{x} = 5$

Step 2

Simplify each value in the list.

Tap for more steps...

Step 2.1

Convert

$1$ to a decimal value.

$1$

Step 2.2

Convert

$2$ to a decimal value.

$2$

Step 2.3

Convert

$3$ to a decimal value.

$3$

Step 2.4

Convert

$4$ to a decimal value.

$4$

Step 2.5

Convert

$5$ to a decimal value.

$5$

Step 2.6

Convert

$6$ to a decimal value.

$6$

Step 2.7

Convert

$7$ to a decimal value.

$7$

Step 2.8

Convert

$8$ to a decimal value.

$8$

Step 2.9

Convert

$9$ to a decimal value.

$9$

Step 2.10

The simplified values are

$1, 2, 3, 4, 5, 6, 7, 8, 9$ .

$1, 2, 3, 4, 5, 6, 7, 8, 9$

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.

$s = \sqrt{\frac{{(1 - 5)}^{2} + {(2 - 5)}^{2} + {(3 - 5)}^{2} + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5

Simplify the result.

Tap for more steps...

Step 5.1

Simplify the expression.

Tap for more steps...

Step 5.1.1

Subtract

$5$ from

$1$ .

$s = \sqrt{\frac{{(- 4)}^{2} + {(2 - 5)}^{2} + {(3 - 5)}^{2} + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.2

Raise

$- 4$ to the power of

$2$ .

$s = \sqrt{\frac{16 + {(2 - 5)}^{2} + {(3 - 5)}^{2} + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.3

Subtract

$5$ from

$2$ .

$s = \sqrt{\frac{16 + {(- 3)}^{2} + {(3 - 5)}^{2} + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.4

Raise

$- 3$ to the power of

$2$ .

$s = \sqrt{\frac{16 + 9 + {(3 - 5)}^{2} + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.5

Subtract

$5$ from

$3$ .

$s = \sqrt{\frac{16 + 9 + {(- 2)}^{2} + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.6

Raise

$- 2$ to the power of

$2$ .

$s = \sqrt{\frac{16 + 9 + 4 + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.7

Subtract

$5$ from

$4$ .

$s = \sqrt{\frac{16 + 9 + 4 + {(- 1)}^{2} + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.8

Raise

$- 1$ to the power of

$2$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + {(5 - 5)}^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.9

Subtract

$5$ from

$5$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0^{2} + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.10

Raising

$0$ to any positive power yields

$0$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + {(6 - 5)}^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.11

Subtract

$5$ from

$6$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1^{2} + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.12

One to any power is one.

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1 + {(7 - 5)}^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.13

Subtract

$5$ from

$7$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1 + 2^{2} + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.14

Raise

$2$ to the power of

$2$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1 + 4 + {(8 - 5)}^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.15

Subtract

$5$ from

$8$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1 + 4 + 3^{2} + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.16

Raise

$3$ to the power of

$2$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + {(9 - 5)}^{2}}{9 - 1}}$

Step 5.1.17

Subtract

$5$ from

$9$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + 4^{2}}{9 - 1}}$

Step 5.1.18

Raise

$4$ to the power of

$2$ .

$s = \sqrt{\frac{16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + 16}{9 - 1}}$

Step 5.1.19

Add

$16$ and

$9$ .

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

Step 5.1.20

Add

$25$ and

$4$ .

$s = \sqrt{\frac{29 + 1 + 0 + 1 + 4 + 9 + 16}{9 - 1}}$

Step 5.1.21

Add

$29$ and

$1$ .

$s = \sqrt{\frac{30 + 0 + 1 + 4 + 9 + 16}{9 - 1}}$

Step 5.1.22

Add

$30$ and

$0$ .

$s = \sqrt{\frac{30 + 1 + 4 + 9 + 16}{9 - 1}}$

Step 5.1.23

Add

$30$ and

$1$ .

$s = \sqrt{\frac{31 + 4 + 9 + 16}{9 - 1}}$

Step 5.1.24

Add

$31$ and

$4$ .

$s = \sqrt{\frac{35 + 9 + 16}{9 - 1}}$

Step 5.1.25

Add

$35$ and

$9$ .

$s = \sqrt{\frac{44 + 16}{9 - 1}}$

Step 5.1.26

Add

$44$ and

$16$ .

$s = \sqrt{\frac{60}{9 - 1}}$

Step 5.1.27

Subtract

$1$ from

$9$ .

$s = \sqrt{\frac{60}{8}}$

Step 5.2

Cancel the common factor of

$60$ and

$8$ .

Tap for more steps...

Step 5.2.1

Factor

$4$ out of

$60$ .

$s = \sqrt{\frac{4 (15)}{8}}$

Step 5.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.1

Factor

$4$ out of

$8$ .

$s = \sqrt{\frac{4 \cdot 15}{4 \cdot 2}}$

Step 5.2.2.2

Cancel the common factor.

$s = \sqrt{\frac{4 \cdot 15}{4 \cdot 2}}$

Step 5.2.2.3

Rewrite the expression.

$s = \sqrt{\frac{15}{2}}$

Step 5.3

Rewrite

$\sqrt{\frac{15}{2}}$ as

$\frac{\sqrt{15}}{\sqrt{2}}$ .

$s = \frac{\sqrt{15}}{\sqrt{2}}$

Step 5.4

Multiply

$\frac{\sqrt{15}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$s = \frac{\sqrt{15}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.5

Combine and simplify the denominator.

Tap for more steps...

Step 5.5.1

Multiply

$\frac{\sqrt{15}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$s = \frac{\sqrt{15} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.5.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$s = \frac{\sqrt{15} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.5.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$s = \frac{\sqrt{15} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.5.4

Use the power rule

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

$s = \frac{\sqrt{15} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.5.5

Add

$1$ and

$1$ .

$s = \frac{\sqrt{15} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.5.6

Rewrite

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

$2$ .

Tap for more steps...

Step 5.5.6.1

Use

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

$\sqrt{2}$ as

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

$s = \frac{\sqrt{15} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.5.6.2

Apply the power rule and multiply exponents,

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

$s = \frac{\sqrt{15} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$s = \frac{\sqrt{15} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.5.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 5.5.6.4.1

Cancel the common factor.

$s = \frac{\sqrt{15} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.5.6.4.2

Rewrite the expression.

$s = \frac{\sqrt{15} \sqrt{2}}{2}$

Step 5.5.6.5

Evaluate the exponent.

$s = \frac{\sqrt{15} \sqrt{2}}{2}$

Step 5.6

Simplify the numerator.

Tap for more steps...

Step 5.6.1

Combine using the product rule for radicals.

$s = \frac{\sqrt{15 \cdot 2}}{2}$

Step 5.6.2

Multiply

$15$ by

$2$ .

$s = \frac{\sqrt{30}}{2}$

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.

$2.7$