Find the Variance 92 , 97 , 53 , 90 , 95 , 98

$92$ ,

$97$ ,

$53$ ,

$90$ ,

$95$ ,

$98$

Step 1

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

$\overline{x} = \frac{92 + 97 + 53 + 90 + 95 + 98}{6}$

Step 2

Simplify the numerator.

Add

$92$ and

$97$ .

$\overline{x} = \frac{189 + 53 + 90 + 95 + 98}{6}$

Add

$189$ and

$53$ .

$\overline{x} = \frac{242 + 90 + 95 + 98}{6}$

Add

$242$ and

$90$ .

$\overline{x} = \frac{332 + 95 + 98}{6}$

Add

$332$ and

$95$ .

$\overline{x} = \frac{427 + 98}{6}$

Add

$427$ and

$98$ .

$\overline{x} = \frac{525}{6}$

Step 3

Cancel the common factor of

$525$ and

$6$ .

Factor

$3$ out of

$525$ .

$\overline{x} = \frac{3 (175)}{6}$

Cancel the common factors.

Factor

$3$ out of

$6$ .

$\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}$

Cancel the common factor.

$\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}$

Rewrite the expression.

$\overline{x} = \frac{175}{2}$

Step 4

Divide.

$\overline{x} = 87.5$

Step 5

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 6

Set up the formula for variance for this set of numbers.

$s = \frac{{(92 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract

$87.5$ from

$92$ .

$s = \frac{{4.5}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise

$4.5$ to the power of

$2$ .

$s = \frac{20.25 + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract

$87.5$ from

$97$ .

$s = \frac{20.25 + {9.5}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise

$9.5$ to the power of

$2$ .

$s = \frac{20.25 + 90.25 + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract

$87.5$ from

$53$ .

$s = \frac{20.25 + 90.25 + {(- 34.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise

$- 34.5$ to the power of

$2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract

$87.5$ from

$90$ .

$s = \frac{20.25 + 90.25 + 1190.25 + {2.5}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise

$2.5$ to the power of

$2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract

$87.5$ from

$95$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + {7.5}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise

$7.5$ to the power of

$2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract

$87.5$ from

$98$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {10.5}^{2}}{6 - 1}$

Raise

$10.5$ to the power of

$2$ .

$s = \frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}$

Add

$20.25$ and

$90.25$ .

$s = \frac{110.5 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}$

Add

$110.5$ and

$1190.25$ .

$s = \frac{1300.75 + 6.25 + 56.25 + 110.25}{6 - 1}$

Add

$1300.75$ and

$6.25$ .

$s = \frac{1307 + 56.25 + 110.25}{6 - 1}$

Add

$1307$ and

$56.25$ .

$s = \frac{1363.25 + 110.25}{6 - 1}$

Add

$1363.25$ and

$110.25$ .

$s = \frac{1473.5}{6 - 1}$

Simplify the expression.

Subtract

$1$ from

$6$ .

$s = \frac{1473.5}{5}$

Divide

$1473.5$ by

$5$ .

$s = 294.7$

Step 8

Approximate the result.

$s^{2} \approx 294.7$

Find the Variance 92 , 97 , 53 , 90 , 95 , 98

Related Questions