Find the Variance 127 135 128 131 133 127 122

$127$ $135$ $128$ $131$ $133$ $127$ $122$

Step 1

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

$\overline{x} = \frac{127 + 135 + 128 + 131 + 133 + 127 + 122}{7}$

Step 2

Simplify the numerator.

Add $127$ and $135$ .

$\overline{x} = \frac{262 + 128 + 131 + 133 + 127 + 122}{7}$

Add $262$ and $128$ .

$\overline{x} = \frac{390 + 131 + 133 + 127 + 122}{7}$

Add $390$ and $131$ .

$\overline{x} = \frac{521 + 133 + 127 + 122}{7}$

Add $521$ and $133$ .

$\overline{x} = \frac{654 + 127 + 122}{7}$

Add $654$ and $127$ .

$\overline{x} = \frac{781 + 122}{7}$

Add $781$ and $122$ .

$\overline{x} = \frac{903}{7}$

Step 3

Divide $903$ by $7$ .

$\overline{x} = 129$

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.

$s = \frac{{(127 - 129)}^{2} + {(135 - 129)}^{2} + {(128 - 129)}^{2} + {(131 - 129)}^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $129$ from $127$ .

$s = \frac{{(- 2)}^{2} + {(135 - 129)}^{2} + {(128 - 129)}^{2} + {(131 - 129)}^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

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

$s = \frac{4 + {(135 - 129)}^{2} + {(128 - 129)}^{2} + {(131 - 129)}^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

Subtract $129$ from $135$ .

$s = \frac{4 + 6^{2} + {(128 - 129)}^{2} + {(131 - 129)}^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

Raise $6$ to the power of $2$ .

$s = \frac{4 + 36 + {(128 - 129)}^{2} + {(131 - 129)}^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

Subtract $129$ from $128$ .

$s = \frac{4 + 36 + {(- 1)}^{2} + {(131 - 129)}^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

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

$s = \frac{4 + 36 + 1 + {(131 - 129)}^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

Subtract $129$ from $131$ .

$s = \frac{4 + 36 + 1 + 2^{2} + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

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

$s = \frac{4 + 36 + 1 + 4 + {(133 - 129)}^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

Subtract $129$ from $133$ .

$s = \frac{4 + 36 + 1 + 4 + 4^{2} + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

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

$s = \frac{4 + 36 + 1 + 4 + 16 + {(127 - 129)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

Subtract $129$ from $127$ .

$s = \frac{4 + 36 + 1 + 4 + 16 + {(- 2)}^{2} + {(122 - 129)}^{2}}{7 - 1}$

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

$s = \frac{4 + 36 + 1 + 4 + 16 + 4 + {(122 - 129)}^{2}}{7 - 1}$

Subtract $129$ from $122$ .

$s = \frac{4 + 36 + 1 + 4 + 16 + 4 + {(- 7)}^{2}}{7 - 1}$

Raise $- 7$ to the power of $2$ .

$s = \frac{4 + 36 + 1 + 4 + 16 + 4 + 49}{7 - 1}$

Add $4$ and $36$ .

$s = \frac{40 + 1 + 4 + 16 + 4 + 49}{7 - 1}$

Add $40$ and $1$ .

$s = \frac{41 + 4 + 16 + 4 + 49}{7 - 1}$

Add $41$ and $4$ .

$s = \frac{45 + 16 + 4 + 49}{7 - 1}$

Add $45$ and $16$ .

$s = \frac{61 + 4 + 49}{7 - 1}$

Add $61$ and $4$ .

$s = \frac{65 + 49}{7 - 1}$

Add $65$ and $49$ .

$s = \frac{114}{7 - 1}$

Simplify the expression.

Subtract $1$ from $7$ .

$s = \frac{114}{6}$

Divide $114$ by $6$ .

$s = 19$

Step 7

Approximate the result.

$s^{2} \approx 19$

Find the Variance 127 135 128 131 133 127 122

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