Find the Variance 197 , 195 , 192 , 203 , 185 , 204 , 182

$197$ ,

$195$ ,

$192$ ,

$203$ ,

$185$ ,

$204$ ,

$182$

Step 1

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

$\overline{x} = \frac{197 + 195 + 192 + 203 + 185 + 204 + 182}{7}$

Step 2

Simplify the numerator.

Add

$197$ and

$195$ .

$\overline{x} = \frac{392 + 192 + 203 + 185 + 204 + 182}{7}$

Add

$392$ and

$192$ .

$\overline{x} = \frac{584 + 203 + 185 + 204 + 182}{7}$

Add

$584$ and

$203$ .

$\overline{x} = \frac{787 + 185 + 204 + 182}{7}$

Add

$787$ and

$185$ .

$\overline{x} = \frac{972 + 204 + 182}{7}$

Add

$972$ and

$204$ .

$\overline{x} = \frac{1176 + 182}{7}$

Add

$1176$ and

$182$ .

$\overline{x} = \frac{1358}{7}$

Step 3

Divide

$1358$ by

$7$ .

$\overline{x} = 194$

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.

$s = \frac{{(197 - 194)}^{2} + {(195 - 194)}^{2} + {(192 - 194)}^{2} + {(203 - 194)}^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract

$194$ from

$197$ .

$s = \frac{3^{2} + {(195 - 194)}^{2} + {(192 - 194)}^{2} + {(203 - 194)}^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Raise

$3$ to the power of

$2$ .

$s = \frac{9 + {(195 - 194)}^{2} + {(192 - 194)}^{2} + {(203 - 194)}^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Subtract

$194$ from

$195$ .

$s = \frac{9 + 1^{2} + {(192 - 194)}^{2} + {(203 - 194)}^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

One to any power is one.

$s = \frac{9 + 1 + {(192 - 194)}^{2} + {(203 - 194)}^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Subtract

$194$ from

$192$ .

$s = \frac{9 + 1 + {(- 2)}^{2} + {(203 - 194)}^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Raise

$- 2$ to the power of

$2$ .

$s = \frac{9 + 1 + 4 + {(203 - 194)}^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Subtract

$194$ from

$203$ .

$s = \frac{9 + 1 + 4 + 9^{2} + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Raise

$9$ to the power of

$2$ .

$s = \frac{9 + 1 + 4 + 81 + {(185 - 194)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Subtract

$194$ from

$185$ .

$s = \frac{9 + 1 + 4 + 81 + {(- 9)}^{2} + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Raise

$- 9$ to the power of

$2$ .

$s = \frac{9 + 1 + 4 + 81 + 81 + {(204 - 194)}^{2} + {(182 - 194)}^{2}}{7 - 1}$

Subtract

$194$ from

$204$ .

$s = \frac{9 + 1 + 4 + 81 + 81 + 10^{2} + {(182 - 194)}^{2}}{7 - 1}$

Raise

$10$ to the power of

$2$ .

$s = \frac{9 + 1 + 4 + 81 + 81 + 100 + {(182 - 194)}^{2}}{7 - 1}$

Subtract

$194$ from

$182$ .

$s = \frac{9 + 1 + 4 + 81 + 81 + 100 + {(- 12)}^{2}}{7 - 1}$

Raise

$- 12$ to the power of

$2$ .

$s = \frac{9 + 1 + 4 + 81 + 81 + 100 + 144}{7 - 1}$

Add

$9$ and

$1$ .

$s = \frac{10 + 4 + 81 + 81 + 100 + 144}{7 - 1}$

Add

$10$ and

$4$ .

$s = \frac{14 + 81 + 81 + 100 + 144}{7 - 1}$

Add

$14$ and

$81$ .

$s = \frac{95 + 81 + 100 + 144}{7 - 1}$

Add

$95$ and

$81$ .

$s = \frac{176 + 100 + 144}{7 - 1}$

Add

$176$ and

$100$ .

$s = \frac{276 + 144}{7 - 1}$

Add

$276$ and

$144$ .

$s = \frac{420}{7 - 1}$

Simplify the expression.

Subtract

$1$ from

$7$ .

$s = \frac{420}{6}$

Divide

$420$ by

$6$ .

$s = 70$

Step 7

Approximate the result.

$s^{2} \approx 70$

Find the Variance 197 , 195 , 192 , 203 , 185 , 204 , 182

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