Comparing Distributions of a Quantitative Variable Chapter Notes | AP Statistics - Grade 9 PDF Download

In statistics, understanding how to describe, summarize, and visualize data distributions is essential. Now, let’s take it a step further by exploring how to compare multiple datasets using various graphical tools like stem-and-leaf plots, histograms, and box plots. This guide, tailored for AP Statistics students, dives into practical examples to sharpen your skills in analyzing and comparing quantitative variables.

Warming Up: Comparing Groups with Stem-and-Leaf Plots

Practice Question

Group N:
1 | 0
2 | 3 6
3 | 5
4 | 1
5 | 4 7
6 | 2

Analysis

• Range: Group M’s weights span from 14 kg to 50 kg, a range of 36 kg. Group N’s weights range from 10 kg to 62 kg, a wider range of 52 kg, indicating greater variability in Group N, with some animals notably heavier than Group M’s heaviest.
• Distribution Patterns: Group M shows a concentration of weights in the 20s and 30s, suggesting a tighter clustering of similar weights. Group N, however, displays a more spread-out distribution across its range, reflecting greater diversity in weights.

Conclusion: Group N exhibits a broader range and more varied weight distribution compared to Group M, which has a more compact and clustered distribution.

AP-Style Practice: Comparing Groups with Histograms

Part A: Estimating the Median
The task is to estimate the median P-T ratio for each group:

• West (n=24): The median lies between the 12th and 13th values, both falling in the 15–16 interval.
• East (n=26): The median lies between the 13th and 14th values, also in the 15–16 interval.

Thus, both groups have a median P-T ratio between 15 and 16 students per teacher.

Part B: Comparing Distributions
Analyze the distributions by examining shape, center, and spread:

• Shape: The West histogram is unimodal and skewed to the right, while the East histogram is unimodal and nearly symmetric, indicating different distribution patterns.
• Center: As established, both medians are approximately 15–16 students per teacher.
• Spread: The West’s range is approximately 22 – 12 = 10, while the East’s is 19 – 12 = 7. The West shows greater variability in P-T ratios compared to the more consistent East.

Part C: Comparing Means
Given the distribution shapes:

• The West’s right skew suggests the mean will exceed the median, pulled higher by larger values in the right tail.
• The East’s near-symmetric distribution implies the mean is close to the median.

Conclusion: The mean P-T ratio for the West is likely higher than for the East due to the right skew in the West’s distribution.

AP-Style Practice: Comparing Groups with Box Plots

Ques: Based on the boxplots, did  basketball players who received visualization training tend to need fewer attempts to make two consecutive baskets from a distance of 22 feet than players who did not receive the training? Explain your reasoning.
Sol: We have two groups, with 10 basketball players randomly assigned to each group. We learn from the question that group 1 received visualization training but group 2. There are a few things here we can compare to find the answer to the question. We can see both groups have the same minimum attempts, and all other measures are different. 
25% of the time the group 1 made the basket in 3 trials but group 2 in 4 trials. 
Now, look at the median. The median is much lower for group 1 than for group 2. Group 1 has an outlier, which is still less than the maximum of group 2. We can see that the training had an impact on group 1, as all the 5 summary measures are less than from group 2; however, we are not asked to generalize this finding yet.  
Finally, to answer the question: it is good enough only to report the median. Because the median number of attempts for players who received visualization training (4) is less than the median number of attempts for players who did not receive training (7), those who received visualization training tend to need fewer attempts to make two consecutive baskets. Talk about talent and persistence!

Key Terms to Understand