To compare the student grades from Mr. Turner’s two Math 2 classes using the specified methods, let’s start with each method step-by-step.

Method 1: 5-Number Summary & Boxplots

Class 1 Grades:

• Data: 72, 80, 79, 81, 75, 82, 77, 80, 86, 42, 77, 88, 78, 80, 77, 74, 90, 84, 81, 86, 88
• Sorted Data: 42, 72, 74, 75, 77, 77, 77, 78, 79, 80, 80, 80, 81, 81, 84, 86, 86, 88, 88, 88, 90

5-Number Summary:

• Minimum: 42
• Q1 (1st Quartile): 77
• Median (Q2): 80
• Q3 (3rd Quartile): 86
• Maximum: 90

Class 2 Grades:

• Data: 84, 86, 86, 88, 88, 70, 89, 88, 84, 90, 82, 77, 86, 86, 61, 70, 88, 87, 86, 79, 84
• Sorted Data: 61, 70, 70, 77, 79, 82, 84, 84, 86, 86, 86, 86, 86, 88, 88, 88, 88, 89, 90

5-Number Summary:

• Minimum: 61
• Q1: 84
• Median: 86
• Q3: 88
• Maximum: 90

Boxplots:

• Create boxplots for both classes using the 5-number summaries. This visual representation will help identify the spread and skewness of the data.

Method 2: Mean and Standard Deviation (Population)

Class 1:

• Mean = (Sum of all grades) / (Number of grades) $\text{Mean} = \frac{72 + 80 + 79 + 81 + 75 + 82 + 77 + 80 + 86 + 42 + 77 + 88 + 78 + 80 + 77 + 74 + 90 + 84 + 81 + 86 + 88}{21} \approx 78.57$
• Standard Deviation (Population): $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \approx 10.94$

Class 2:

• Mean: $\text{Mean} = \frac{84 + 86 + 86 + 88 + 88 + 70 + 89 + 88 + 84 + 90 + 82 + 77 + 86 + 86 + 61 + 70 + 88 + 87 + 86 + 79 + 84}{21} \approx 82.57$
• Standard Deviation: $\sigma \approx 7.82$

Method 3: Histograms

To create histograms for both classes:

• Divide the grade range into intervals (bins) and count the number of grades that fall into each bin.
• Plot the histograms for visual comparison of the distributions.

Method 4: Dot Plots

Create dot plots for both classes:

• Each grade will be represented as a dot along a number line. This will provide a clear view of the frequency of each grade.

Interpretation of Results

1. 5-Number Summary & Boxplots: Class 1 has a wider spread and a lower minimum score, suggesting a greater variation in performance. Class 2 has a higher median and quartiles, indicating generally better performance.

2. Mean and Standard Deviation: Class 1 has a lower mean and higher standard deviation compared to Class 2, indicating that while Class 1 has some low scores, Class 2 shows more consistent performance.

3. Histograms and Dot Plots: The shape of the distributions can be assessed for skewness and modality (unimodal vs. bimodal), which can indicate how well the teaching methods worked for each group.

Would you like more detailed calculations for any of these methods, or do you have specific questions? Here are some follow-up questions to consider:

1. How would you interpret the impact of teaching methods on student performance based on the results?
2. Would you like to visualize the histograms or dot plots?
3. How do outliers affect the overall analysis of these datasets?
4. What additional statistical methods could be applied for deeper insights?
5. How could you apply these findings to improve teaching strategies in the future?

Tip: Always consider the context of the data and potential external factors that might influence the results when analyzing student performance.