AP Statistics: Exploring one variable data
Understanding and comparing distributions is a cornerstone of data analysis. By comparing datasets, we uncover patterns, trends, and insights that are otherwise hidden. Let’s explore how to compare distributions using three popular graphical representations: stem-and-leaf plots, histograms, and box plots.
Comparing Groups with Stem-and-Leaf Plots
Warm-Up ExampleThe weights of two groups of animals, Group M and Group N, are recorded in the following stem-and-leaf plots (weights in kilograms):
Group M:
1 | 4
2 | 3 4 8
3 | 2 6 8
4 |
5 | 0
Group N:
1 | 0
2 | 3 6
3 | 5
4 | 1
5 | 4 7
6 | 2
To compare:
Range:
Group M: 14–50 kg
Group N: 10–62 kg
Group N has a wider range of weights.
Distribution:
Group M clusters around the 20s and 30s, suggesting more uniform weights.
Group N exhibits a more diverse spread.
Conclusion:Group N’s wider range and varied distribution indicate a greater diversity in weight compared to Group M.
Comparing Groups with Histograms
AP-Style Problem
Consider the pupil-to-teacher (P-T) ratio for states west and east of the Mississippi River during the 2001–2002 school year. Two histograms show the distribution of ratios for the 24 western states and 26 eastern states.
Part (a): Estimate the Median
For western states (n=24n = 24), the median falls between the 12th and 13th values, both in the 15–16 interval.
For eastern states (n=26n = 26), the median falls between the 13th and 14th values, also in the 15–16 interval.
Both groups share a similar median: 15–16 students per teacher.
Part (b): Compare Shape, Center, and Spread
Shape:
West: Unimodal and skewed right.
East: Unimodal and symmetric.
Center:
Both have a median in the 15–16 interval.
Spread:
West: Range ≈ 22−12=1022 - 12 = 10.
East: Range ≈ 19−12=719 - 12 = 7.
The West has a broader range, indicating more variability.
Part (c): Comparing Means
West: The skewed-right distribution implies a mean greater than the median.
East: The symmetric distribution suggests the mean is close to the median.
Conclusion: The mean P-T ratio for western states is likely higher than for eastern states.
Comparing Groups with Box Plots
AP-Style Problem: Visualization Training in Basketball
A study investigated whether visualization techniques impact basketball performance. Two groups of players attempted to make two consecutive baskets. The number of attempts is summarized in the box plots below.
Insights:
Minimum Values: Both groups share the same minimum attempts.
Quartiles:
Group 1 (with training): Q1 = 3; Median = 4.
Group 2 (without training): Q1 = 4; Median = 7.
Outliers: Group 1 has an outlier, but it is still lower than Group 2’s maximum.
Conclusion:
The lower median for Group 1 suggests players with visualization training needed fewer attempts to succeed. This highlights the potential impact of the training.
Key Takeaways
Stem-and-Leaf Plots offer detailed views of individual data points, enabling direct comparisons of ranges and clusters.
Histograms reveal distribution shapes, medians, and variability, crucial for understanding overall trends.
Box Plots provide a concise summary of five-number statistics, making it easy to compare central tendencies and spreads between groups.
When comparing distributions, always examine shape, center, and spread, as they provide the clearest insights into the underlying patterns of your data.
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