https://file.aiquickdraw.com/mathbot/file/1726250428079vea243oq.jpg

The problem shown in the image is asking for the variance of a set of test scores, specifically focusing on finding the numerator of the variance calculation.

### Given Data:
Test scores are:  
\[ 90, 75, 72, 88, 85 \]

### Formula for Variance (Population Variance):
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
Where:
- \( x_i \) represents each data point (the test scores),
- \( \mu \) is the mean of the data set,
- \( N \) is the total number of data points,
- The numerator of the formula is \( \sum (x_i - \mu)^2 \), which sums the squared differences from the mean.

### Step 1: Calculate the Mean (\( \mu \))  
\[
\mu = \frac{90 + 75 + 72 + 88 + 85}{5} = \frac{410}{5} = 82
\]

### Step 2: Calculate the Squared Differences  
For each score \( x_i \):
\[
(90 - 82)^2 = 64
\]
\[
(75 - 82)^2 = 49
\]
\[
(72 - 82)^2 = 100
\]
\[
(88 - 82)^2 = 36
\]
\[
(85 - 82)^2 = 9
\]

### Step 3: Sum of Squared Differences  
\[
64 + 49 + 100 + 36 + 9 = 258
\]

### Step 4: Conclusion  
The value of the numerator \( \sum (x_i - \mu)^2 \) is **258**, which corresponds to the correct answer.

The answer is **258**.

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How do you calculate variance for a sample instead of a population?
2. Why is it necessary to square the differences when calculating variance?
3. What is the significance of variance in statistical analysis?
4. How do you compute the standard deviation from variance?
5. How would the calculation change if you had an additional test score?

**Tip:** When dealing with population variance, remember that every data point is considered without adjusting for sample size.

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?

Learn how to calculate the variance for a class population based on the test scores: 90, 75, 72, 88, and 85. This example covers key statistical concepts, including finding the mean and summing the squared differences. Suitable for Grades 9-11.

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