Find the Variance 82 , 76 , 93 , 82 , 73 , 80

$82$ , $76$ , $93$ , $82$ , $73$ , $80$

Step 1

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

$\overline{x} = \frac{82 + 76 + 93 + 82 + 73 + 80}{6}$

Step 2

Simplify the numerator.

Add $82$ and $76$ .

$\overline{x} = \frac{158 + 93 + 82 + 73 + 80}{6}$

Add $158$ and $93$ .

$\overline{x} = \frac{251 + 82 + 73 + 80}{6}$

Add $251$ and $82$ .

$\overline{x} = \frac{333 + 73 + 80}{6}$

Add $333$ and $73$ .

$\overline{x} = \frac{406 + 80}{6}$

Add $406$ and $80$ .

$\overline{x} = \frac{486}{6}$

Step 3

Divide $486$ by $6$ .

$\overline{x} = 81$

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.

$s = \frac{{(82 - 81)}^{2} + {(76 - 81)}^{2} + {(93 - 81)}^{2} + {(82 - 81)}^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $81$ from $82$ .

$s = \frac{1^{2} + {(76 - 81)}^{2} + {(93 - 81)}^{2} + {(82 - 81)}^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

One to any power is one.

$s = \frac{1 + {(76 - 81)}^{2} + {(93 - 81)}^{2} + {(82 - 81)}^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Subtract $81$ from $76$ .

$s = \frac{1 + {(- 5)}^{2} + {(93 - 81)}^{2} + {(82 - 81)}^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Raise $- 5$ to the power of $2$ .

$s = \frac{1 + 25 + {(93 - 81)}^{2} + {(82 - 81)}^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Subtract $81$ from $93$ .

$s = \frac{1 + 25 + 12^{2} + {(82 - 81)}^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Raise $12$ to the power of $2$ .

$s = \frac{1 + 25 + 144 + {(82 - 81)}^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Subtract $81$ from $82$ .

$s = \frac{1 + 25 + 144 + 1^{2} + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

One to any power is one.

$s = \frac{1 + 25 + 144 + 1 + {(73 - 81)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Subtract $81$ from $73$ .

$s = \frac{1 + 25 + 144 + 1 + {(- 8)}^{2} + {(80 - 81)}^{2}}{6 - 1}$

Raise $- 8$ to the power of $2$ .

$s = \frac{1 + 25 + 144 + 1 + 64 + {(80 - 81)}^{2}}{6 - 1}$

Subtract $81$ from $80$ .

$s = \frac{1 + 25 + 144 + 1 + 64 + {(- 1)}^{2}}{6 - 1}$

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

$s = \frac{1 + 25 + 144 + 1 + 64 + 1}{6 - 1}$

Add $1$ and $25$ .

$s = \frac{26 + 144 + 1 + 64 + 1}{6 - 1}$

Add $26$ and $144$ .

$s = \frac{170 + 1 + 64 + 1}{6 - 1}$

Add $170$ and $1$ .

$s = \frac{171 + 64 + 1}{6 - 1}$

Add $171$ and $64$ .

$s = \frac{235 + 1}{6 - 1}$

Add $235$ and $1$ .

$s = \frac{236}{6 - 1}$

Subtract $1$ from $6$ .

$s = \frac{236}{5}$

Step 7

Approximate the result.

$s^{2} \approx 47.2$

Find the Variance 82 , 76 , 93 , 82 , 73 , 80

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