Basic Math Examples

$92$ , $97$ , $53$ , $90$ , $95$ , $98$

Step 1

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

$\overline{x} = \frac{92 + 97 + 53 + 90 + 95 + 98}{6}$

Step 2

Simplify the numerator.

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Step 2.1

Add $92$ and $97$ .

$\overline{x} = \frac{189 + 53 + 90 + 95 + 98}{6}$

Step 2.2

Add $189$ and $53$ .

$\overline{x} = \frac{242 + 90 + 95 + 98}{6}$

Step 2.3

Add $242$ and $90$ .

$\overline{x} = \frac{332 + 95 + 98}{6}$

Step 2.4

Add $332$ and $95$ .

$\overline{x} = \frac{427 + 98}{6}$

Step 2.5

Add $427$ and $98$ .

$\overline{x} = \frac{525}{6}$

Step 3

Cancel the common factor of $525$ and $6$ .

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Step 3.1

Factor $3$ out of $525$ .

$\overline{x} = \frac{3 (175)}{6}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $6$ .

$\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}$

Step 3.2.2

Cancel the common factor.

$\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}$

Step 3.2.3

Rewrite the expression.

$\overline{x} = \frac{175}{2}$

Step 4

Divide.

$\overline{x} = 87.5$

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.

$s = \frac{{(92 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $87.5$ from $92$ .

$s = \frac{{4.5}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.2

Raise $4.5$ to the power of $2$ .

$s = \frac{20.25 + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.3

Subtract $87.5$ from $97$ .

$s = \frac{20.25 + {9.5}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.4

Raise $9.5$ to the power of $2$ .

$s = \frac{20.25 + 90.25 + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.5

Subtract $87.5$ from $53$ .

$s = \frac{20.25 + 90.25 + {(- 34.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.6

Raise $- 34.5$ to the power of $2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.7

Subtract $87.5$ from $90$ .

$s = \frac{20.25 + 90.25 + 1190.25 + {2.5}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.8

Raise $2.5$ to the power of $2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.9

Subtract $87.5$ from $95$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + {7.5}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.10

Raise $7.5$ to the power of $2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7.1.11

Subtract $87.5$ from $98$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {10.5}^{2}}{6 - 1}$

Step 7.1.12

Raise $10.5$ to the power of $2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}$

Step 7.1.13

Add $20.25$ and $90.25$ .

$s = \frac{110.5 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}$

Step 7.1.14

Add $110.5$ and $1190.25$ .

$s = \frac{1300.75 + 6.25 + 56.25 + 110.25}{6 - 1}$

Step 7.1.15

Add $1300.75$ and $6.25$ .

$s = \frac{1307 + 56.25 + 110.25}{6 - 1}$

Step 7.1.16

Add $1307$ and $56.25$ .

$s = \frac{1363.25 + 110.25}{6 - 1}$

Step 7.1.17

Add $1363.25$ and $110.25$ .

$s = \frac{1473.5}{6 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $6$ .

$s = \frac{1473.5}{5}$

Step 7.2.2

Divide $1473.5$ by $5$ .

$s = 294.7$

Step 8

Approximate the result.

$s^{2} \approx 294.7$