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

$53$ , $90$ , $92$ , $95$ , $97$ , $98$

Step 1

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

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

Step 2

Simplify the numerator.

Add $53$ and $90$ .

$\overline{x} = \frac{143 + 92 + 95 + 97 + 98}{6}$

Add $143$ and $92$ .

$\overline{x} = \frac{235 + 95 + 97 + 98}{6}$

Add $235$ and $95$ .

$\overline{x} = \frac{330 + 97 + 98}{6}$

Add $330$ and $97$ .

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

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.

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

Step 7

Simplify the result.

Simplify the numerator.

Subtract $87.5$ from $53$ .

$s = \frac{{(- 34.5)}^{2} + {(90 - 87.5)}^{2} + {(92 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise $- 34.5$ to the power of $2$ .

$s = \frac{1190.25 + {(90 - 87.5)}^{2} + {(92 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract $87.5$ from $90$ .

$s = \frac{1190.25 + {2.5}^{2} + {(92 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise $2.5$ to the power of $2$ .

$s = \frac{1190.25 + 6.25 + {(92 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract $87.5$ from $92$ .

$s = \frac{1190.25 + 6.25 + {4.5}^{2} + {(95 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise $4.5$ to the power of $2$ .

$s = \frac{1190.25 + 6.25 + 20.25 + {(95 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract $87.5$ from $95$ .

$s = \frac{1190.25 + 6.25 + 20.25 + {7.5}^{2} + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise $7.5$ to the power of $2$ .

$s = \frac{1190.25 + 6.25 + 20.25 + 56.25 + {(97 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Subtract $87.5$ from $97$ .

$s = \frac{1190.25 + 6.25 + 20.25 + 56.25 + {9.5}^{2} + {(98 - 87.5)}^{2}}{6 - 1}$

Raise $9.5$ to the power of $2$ .

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

Subtract $87.5$ from $98$ .

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

Raise $10.5$ to the power of $2$ .

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

Add $1190.25$ and $6.25$ .

$s = \frac{1196.5 + 20.25 + 56.25 + 90.25 + 110.25}{6 - 1}$

Add $1196.5$ and $20.25$ .

$s = \frac{1216.75 + 56.25 + 90.25 + 110.25}{6 - 1}$

Add $1216.75$ and $56.25$ .

$s = \frac{1273 + 90.25 + 110.25}{6 - 1}$

Add $1273$ and $90.25$ .

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

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 53 , 90 , 92 , 95 , 97 , 98

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