Find the Variance 4 4 7 7

$4$ $4$ $7$ $7$

Step 1

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

$\overline{x} = \frac{4 + 4 + 7 + 7}{4}$

Step 2

Simplify the numerator.

Add $4$ and $4$ .

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

Add $8$ and $7$ .

$\overline{x} = \frac{15 + 7}{4}$

Add $15$ and $7$ .

$\overline{x} = \frac{22}{4}$

Step 3

Cancel the common factor of $22$ and $4$ .

Factor $2$ out of $22$ .

$\overline{x} = \frac{2 (11)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\overline{x} = \frac{2 \cdot 11}{2 \cdot 2}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 11}{2 \cdot 2}$

Rewrite the expression.

$\overline{x} = \frac{11}{2}$

Step 4

Divide.

$\overline{x} = 5.5$

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.

$s = \frac{{(4 - 5.5)}^{2} + {(4 - 5.5)}^{2} + {(7 - 5.5)}^{2} + {(7 - 5.5)}^{2}}{4 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract $5.5$ from $4$ .

$s = \frac{{(- 1.5)}^{2} + {(4 - 5.5)}^{2} + {(7 - 5.5)}^{2} + {(7 - 5.5)}^{2}}{4 - 1}$

Raise $- 1.5$ to the power of $2$ .

$s = \frac{2.25 + {(4 - 5.5)}^{2} + {(7 - 5.5)}^{2} + {(7 - 5.5)}^{2}}{4 - 1}$

Subtract $5.5$ from $4$ .

$s = \frac{2.25 + {(- 1.5)}^{2} + {(7 - 5.5)}^{2} + {(7 - 5.5)}^{2}}{4 - 1}$

Raise $- 1.5$ to the power of $2$ .

$s = \frac{2.25 + 2.25 + {(7 - 5.5)}^{2} + {(7 - 5.5)}^{2}}{4 - 1}$

Subtract $5.5$ from $7$ .

$s = \frac{2.25 + 2.25 + {1.5}^{2} + {(7 - 5.5)}^{2}}{4 - 1}$

Raise $1.5$ to the power of $2$ .

$s = \frac{2.25 + 2.25 + 2.25 + {(7 - 5.5)}^{2}}{4 - 1}$

Subtract $5.5$ from $7$ .

$s = \frac{2.25 + 2.25 + 2.25 + {1.5}^{2}}{4 - 1}$

Raise $1.5$ to the power of $2$ .

$s = \frac{2.25 + 2.25 + 2.25 + 2.25}{4 - 1}$

Add $2.25$ and $2.25$ .

$s = \frac{4.5 + 2.25 + 2.25}{4 - 1}$

Add $4.5$ and $2.25$ .

$s = \frac{6.75 + 2.25}{4 - 1}$

Add $6.75$ and $2.25$ .

$s = \frac{9}{4 - 1}$

Simplify the expression.

Subtract $1$ from $4$ .

$s = \frac{9}{3}$

Divide $9$ by $3$ .

$s = 3$

Step 8

Approximate the result.

$s^{2} \approx 3$

Find the Variance 4 4 7 7

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