Question

the test scores of a geometry class are given below.
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
• -6
• 16
• 258

Explanation:

Step1: Calculate the mean (μ)

First, find the mean of the scores 90, 75, 72, 88, 85. The formula for the mean of a population is $\mu=\frac{\sum_{i = 1}^{N}x_{i}}{N}$, where $N = 5$ (number of scores) and $x_{i}$ are the individual scores.
$\sum_{i=1}^{5}x_{i}=90 + 75+72 + 88+85=410$
$\mu=\frac{410}{5}=82$

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 (numerator)

Sum these squared differences: $64 + 49+100 + 36+9=258$

Answer:

258