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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