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Calculating the mean allows us to find the average value of a data set, showing us the typical outcome. To understand this fully, let's look at the context of Ms. Hall's class quiz. Here, the mean of the number of questions students answered correctly, denoted as \(X\), was given to be 7.6. This means that on average, students correctly answered 7.6 out of 10 questions given.

Now, to convert this average number of correct answers into a grade out of 100, Ms. Hall multiplies it by 10, as each correct answer equals 10 points. Therefore, the mean grade \(G\) for the class is calculated by multiplying the mean of \(X\) by 10:

• \(\text{Mean of } G = 10 \times \text{Mean of } X = 10 \times 7.6 = 76\)

This shows that on average, a student scores 76 out of 100 on the quiz.

Standard deviation gives us an insight into the spread or variability of our data points relative to the mean. When it comes to Ms. Hall's quiz scores, the standard deviation of the number of correct answers \(X\) is given as 1.32. This tells us that most students' scores are within 1.32 questions of the mean score, 7.6.

In the context of grades \(G\), since the grade is a linear transformation of the number of correct answers (each correct answer increases the grade by 10 points), the standard deviation of \(G\) is also scaled by this factor:

• \(\text{Standard deviation of } G = 10 \times \text{Standard deviation of } X = 10 \times 1.32 = 13.2\)

This means that the grades deviate, on average, 13.2 points from the mean grade of 76. The larger standard deviation reflects the increased spread of grades due to scaling.

Variance provides another measure of data dispersion, similar to standard deviation, but it represents the average of the squared deviations from the mean.

For \(X\), the variance is the square of its standard deviation:

• \(\text{Variance of } X = (1.32)^2 = 1.7424\)

This indicates the average of the squared deviations from the mean number of correct answers.

Conversely, the variance of \(G\) is the square of its standard deviation, reflecting the spread of grades around their mean:

• \(\text{Variance of } G = (13.2)^2 = 174.24\)

The variance of \(G\) is 100 times that of \(X\). This result occurs because when scaling a data set by a constant factor, the variance changes by the square of that factor. In this case, since the grades are scaled by a factor of 10, the variance therefore increases by \(10^2 = 100\), demonstrating a much bigger spread of the grade data compared to the original number of correct answers.