Find the Variance 90 , 90 , 91 , 89 , 91 , 89 , 90 , 90

$90$ , $90$ , $91$ , $89$ , $91$ , $89$ , $90$ , $90$

Step 1

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

$\overline{x} = \frac{90 + 90 + 91 + 89 + 91 + 89 + 90 + 90}{8}$

Step 2

Simplify the numerator.

Add $90$ and $90$ .

$\overline{x} = \frac{180 + 91 + 89 + 91 + 89 + 90 + 90}{8}$

Add $180$ and $91$ .

$\overline{x} = \frac{271 + 89 + 91 + 89 + 90 + 90}{8}$

Add $271$ and $89$ .

$\overline{x} = \frac{360 + 91 + 89 + 90 + 90}{8}$

Add $360$ and $91$ .

$\overline{x} = \frac{451 + 89 + 90 + 90}{8}$

Add $451$ and $89$ .

$\overline{x} = \frac{540 + 90 + 90}{8}$

Add $540$ and $90$ .

$\overline{x} = \frac{630 + 90}{8}$

Add $630$ and $90$ .

$\overline{x} = \frac{720}{8}$

Step 3

Divide $720$ by $8$ .

$\overline{x} = 90$

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.

$s = \frac{{(90 - 90)}^{2} + {(90 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $90$ from $90$ .

$s = \frac{0^{2} + {(90 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Raising $0$ to any positive power yields $0$ .

$s = \frac{0 + {(90 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Subtract $90$ from $90$ .

$s = \frac{0 + 0^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Raising $0$ to any positive power yields $0$ .

$s = \frac{0 + 0 + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Subtract $90$ from $91$ .

$s = \frac{0 + 0 + 1^{2} + {(89 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

One to any power is one.

$s = \frac{0 + 0 + 1 + {(89 - 90)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Subtract $90$ from $89$ .

$s = \frac{0 + 0 + 1 + {(- 1)}^{2} + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

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

$s = \frac{0 + 0 + 1 + 1 + {(91 - 90)}^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Subtract $90$ from $91$ .

$s = \frac{0 + 0 + 1 + 1 + 1^{2} + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

One to any power is one.

$s = \frac{0 + 0 + 1 + 1 + 1 + {(89 - 90)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Subtract $90$ from $89$ .

$s = \frac{0 + 0 + 1 + 1 + 1 + {(- 1)}^{2} + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

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

$s = \frac{0 + 0 + 1 + 1 + 1 + 1 + {(90 - 90)}^{2} + {(90 - 90)}^{2}}{8 - 1}$

Subtract $90$ from $90$ .

$s = \frac{0 + 0 + 1 + 1 + 1 + 1 + 0^{2} + {(90 - 90)}^{2}}{8 - 1}$

Raising $0$ to any positive power yields $0$ .

$s = \frac{0 + 0 + 1 + 1 + 1 + 1 + 0 + {(90 - 90)}^{2}}{8 - 1}$

Subtract $90$ from $90$ .

$s = \frac{0 + 0 + 1 + 1 + 1 + 1 + 0 + 0^{2}}{8 - 1}$

Raising $0$ to any positive power yields $0$ .

$s = \frac{0 + 0 + 1 + 1 + 1 + 1 + 0 + 0}{8 - 1}$

Add $0 + 0 + 1 + 1 + 1 + 1 + 0$ and $0$ .

$s = \frac{0 + 0 + 1 + 1 + 1 + 1 + 0}{8 - 1}$

Add $0 + 0 + 1 + 1 + 1 + 1$ and $0$ .

$s = \frac{0 + 0 + 1 + 1 + 1 + 1}{8 - 1}$

Add $0$ and $0$ .

$s = \frac{0 + 1 + 1 + 1 + 1}{8 - 1}$

Add $0$ and $1$ .

$s = \frac{1 + 1 + 1 + 1}{8 - 1}$

Add $1$ and $1$ .

$s = \frac{2 + 1 + 1}{8 - 1}$

Add $2$ and $1$ .

$s = \frac{3 + 1}{8 - 1}$

Add $3$ and $1$ .

$s = \frac{4}{8 - 1}$

Subtract $1$ from $8$ .

$s = \frac{4}{7}$

Step 7

Approximate the result.

$s^{2} \approx 0.5714$

Find the Variance 90 , 90 , 91 , 89 , 91 , 89 , 90 , 90

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