Question

the test scores of a geometry class are given.
90, 75, 72, 88, 85
the teacher wants to find the variance for the class population. what is the value of the numerator of the calculation of the variance?
variance: $sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}$
-160 258
-6 16

Explanation:

Step1: Calculate the mean (μ)

First, find the mean of the scores. The scores are 90, 75, 72, 88, 85. The sum of the scores is \( 90 + 75 + 72 + 88 + 85 = 410 \). There are 5 scores, so the mean \( \mu = \frac{410}{5} = 82 \).

Step2: Calculate each \((x_i - \mu)^2\)

• For \( x_1 = 90 \): \( (90 - 82)^2 = 8^2 = 64 \)
• For \( x_2 = 75 \): \( (75 - 82)^2 = (-7)^2 = 49 \)
• For \( x_3 = 72 \): \( (72 - 82)^2 = (-10)^2 = 100 \)
• For \( x_4 = 88 \): \( (88 - 82)^2 = 6^2 = 36 \)
• For \( x_5 = 85 \): \( (85 - 82)^2 = 3^2 = 9 \)

Step3: Sum the squared differences

Now, sum these squared differences: \( 64 + 49 + 100 + 36 + 9 = 258 \). This sum is the numerator of the variance formula.

Answer:

258