Find the Variance 25 , 21 , 26 , 20 , 18

$25$ , $21$ , $26$ , $20$ , $18$

Step 1

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

$\overline{x} = \frac{25 + 21 + 26 + 20 + 18}{5}$

Step 2

Simplify the numerator.

Add $25$ and $21$ .

$\overline{x} = \frac{46 + 26 + 20 + 18}{5}$

Add $46$ and $26$ .

$\overline{x} = \frac{72 + 20 + 18}{5}$

Add $72$ and $20$ .

$\overline{x} = \frac{92 + 18}{5}$

Add $92$ and $18$ .

$\overline{x} = \frac{110}{5}$

Step 3

Divide $110$ by $5$ .

$\overline{x} = 22$

Step 4

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 5

Set up the formula for variance for this set of numbers.

$s = \frac{{(25 - 22)}^{2} + {(21 - 22)}^{2} + {(26 - 22)}^{2} + {(20 - 22)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $22$ from $25$ .

$s = \frac{3^{2} + {(21 - 22)}^{2} + {(26 - 22)}^{2} + {(20 - 22)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

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

$s = \frac{9 + {(21 - 22)}^{2} + {(26 - 22)}^{2} + {(20 - 22)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

Subtract $22$ from $21$ .

$s = \frac{9 + {(- 1)}^{2} + {(26 - 22)}^{2} + {(20 - 22)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

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

$s = \frac{9 + 1 + {(26 - 22)}^{2} + {(20 - 22)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

Subtract $22$ from $26$ .

$s = \frac{9 + 1 + 4^{2} + {(20 - 22)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

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

$s = \frac{9 + 1 + 16 + {(20 - 22)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

Subtract $22$ from $20$ .

$s = \frac{9 + 1 + 16 + {(- 2)}^{2} + {(18 - 22)}^{2}}{5 - 1}$

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

$s = \frac{9 + 1 + 16 + 4 + {(18 - 22)}^{2}}{5 - 1}$

Subtract $22$ from $18$ .

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

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

$s = \frac{9 + 1 + 16 + 4 + 16}{5 - 1}$

Add $9$ and $1$ .

$s = \frac{10 + 16 + 4 + 16}{5 - 1}$

Add $10$ and $16$ .

$s = \frac{26 + 4 + 16}{5 - 1}$

Add $26$ and $4$ .

$s = \frac{30 + 16}{5 - 1}$

Add $30$ and $16$ .

$s = \frac{46}{5 - 1}$

Reduce the expression by cancelling the common factors.

Subtract $1$ from $5$ .

$s = \frac{46}{4}$

Cancel the common factor of $46$ and $4$ .

Factor $2$ out of $46$ .

$s = \frac{2 (23)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$s = \frac{2 \cdot 23}{2 \cdot 2}$

Cancel the common factor.

$s = \frac{2 \cdot 23}{2 \cdot 2}$

Rewrite the expression.

$s = \frac{23}{2}$

Step 7

Approximate the result.

$s^{2} \approx 11.5$

Find the Variance 25 , 21 , 26 , 20 , 18

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