Find the Variance 85 , 65 , 72 , 73 , 53

$85$ , $65$ , $72$ , $73$ , $53$

Step 1

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

$\overline{x} = \frac{85 + 65 + 72 + 73 + 53}{5}$

Step 2

Simplify the numerator.

Add $85$ and $65$ .

$\overline{x} = \frac{150 + 72 + 73 + 53}{5}$

Add $150$ and $72$ .

$\overline{x} = \frac{222 + 73 + 53}{5}$

Add $222$ and $73$ .

$\overline{x} = \frac{295 + 53}{5}$

Add $295$ and $53$ .

$\overline{x} = \frac{348}{5}$

Step 3

Divide.

$\overline{x} = 69.6$

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.

$s = \frac{{(85 - 69.6)}^{2} + {(65 - 69.6)}^{2} + {(72 - 69.6)}^{2} + {(73 - 69.6)}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $69.6$ from $85$ .

$s = \frac{{15.4}^{2} + {(65 - 69.6)}^{2} + {(72 - 69.6)}^{2} + {(73 - 69.6)}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Raise $15.4$ to the power of $2$ .

$s = \frac{237.16 + {(65 - 69.6)}^{2} + {(72 - 69.6)}^{2} + {(73 - 69.6)}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Subtract $69.6$ from $65$ .

$s = \frac{237.16 + {(- 4.6)}^{2} + {(72 - 69.6)}^{2} + {(73 - 69.6)}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Raise $- 4.6$ to the power of $2$ .

$s = \frac{237.16 + 21.16 + {(72 - 69.6)}^{2} + {(73 - 69.6)}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Subtract $69.6$ from $72$ .

$s = \frac{237.16 + 21.16 + {2.4}^{2} + {(73 - 69.6)}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Raise $2.4$ to the power of $2$ .

$s = \frac{237.16 + 21.16 + 5.76 + {(73 - 69.6)}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Subtract $69.6$ from $73$ .

$s = \frac{237.16 + 21.16 + 5.76 + {3.4}^{2} + {(53 - 69.6)}^{2}}{5 - 1}$

Raise $3.4$ to the power of $2$ .

$s = \frac{237.16 + 21.16 + 5.76 + 11.56 + {(53 - 69.6)}^{2}}{5 - 1}$

Subtract $69.6$ from $53$ .

$s = \frac{237.16 + 21.16 + 5.76 + 11.56 + {(- 16.6)}^{2}}{5 - 1}$

Raise $- 16.6$ to the power of $2$ .

$s = \frac{237.16 + 21.16 + 5.76 + 11.56 + 275.56}{5 - 1}$

Add $237.16$ and $21.16$ .

$s = \frac{258.32 + 5.76 + 11.56 + 275.56}{5 - 1}$

Add $258.32$ and $5.76$ .

$s = \frac{264.08 + 11.56 + 275.56}{5 - 1}$

Add $264.08$ and $11.56$ .

$s = \frac{275.64 + 275.56}{5 - 1}$

Add $275.64$ and $275.56$ .

$s = \frac{551.2}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$s = \frac{551.2}{4}$

Divide $551.2$ by $4$ .

$s = 137.8$

Step 7

Approximate the result.

$s^{2} \approx 137.8$

Find the Variance 85 , 65 , 72 , 73 , 53

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