1.9 Comparing Distributions of a Quantitative Variable

Comparing distributions means describing two or more groups of quantitative data using the same features and the same vocabulary. Always compare shape, center, and variability, plus any unusual features like outliers, gaps, or clusters, and always do it in context with units.

Why This Matters for the AP Statistics Exam

This skill shows up whenever you see two or more groups shown with side-by-side boxplots, comparative histograms, stacked dotplots, or back-to-back stem plots. You will need to read the graphs or summary statistics and write a clear comparison in context.

The most common mistake here is describing each group on its own instead of comparing them. AP questions in this topic want you to connect the groups directly: which one has a higher center, which one has more spread, which one is more skewed. On free response, writing in context and using true comparison words is important for clear, complete work.

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Key Takeaways

• Compare distributions on shape, center, and variability, plus unusual features (outliers, gaps, clusters, multiple peaks).
• Use comparison words like "greater than," "less than," or "about the same." Do not just describe each group separately.
• Any graph type works for comparing groups: histograms, side-by-side boxplots, dotplots, or stem plots, as long as the groups use the same scale.
• Any numerical summary works too: mean, median, standard deviation, IQR, range, or relative frequency.
• Use resistant measures (median, IQR) when a distribution is skewed or has outliers, and report center and spread together.
• Always include context and units. Numbers without context are incomplete answers.

How to Use This on the AP Statistics Exam

Free Response

When a prompt says "compare the distributions," hit all of these:

• Shape: Is each group symmetric, skewed right, or skewed left? Unimodal or bimodal?
• Center: Which group has a higher median or mean? Or are they about the same?
• Variability: Which group is more spread out? Compare range, IQR, or standard deviation.
• Unusual features: Point out outliers, gaps, or clusters in either group.

Phrase every point as a comparison. "Group A is skewed right while Group B is roughly symmetric" earns more than two separate descriptions.

Choosing Summary Statistics

• For a skewed distribution or one with outliers, the median and IQR describe center and spread better because they are resistant.
• For a roughly symmetric distribution, the mean and standard deviation work well.
• If one group is skewed and the other is symmetric, expect the mean to sit toward the long tail in the skewed group. That can shift how the means compare even when medians look similar.

Common Trap

Reporting only the center is incomplete. A response that compares medians but ignores shape and spread will lose credit. Hit every feature.

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

Try a quick warm up using a familiar graph: the stem plot.

Question: The weights of two groups of eight animals, Group M and Group N, are recorded and the data are shown in the stem plots below (each stem and leaf represents weight in kg). Use the stem plots to compare the weights of the animals in the two groups.

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, look at center, spread, and shape for both groups. Group M has weights from 14 to 50 kg, and Group N has weights from 10 to 62 kg, so Group N has a wider range. Group N has heavier animals at the top than anything in Group M.

Look at how the values cluster too. Group M has most of its values bunched in the 20s and 30s, while Group N spreads more evenly across the whole range. That means Group M's animals are more similar in weight to each other, while Group N's weights are more varied.

Overall, Group N has greater variability than Group M, with a more spread-out distribution.

Comparing Groups with Histograms: Practice AP-Style Problem

Each state in the United States keeps records on the number of pupils enrolled in public schools and the number of teachers employed by public schools for each school year. From these records, the ratio of pupils to teachers (P-T ratio) can be calculated for each state. The histograms below show the P-T ratio for every state during the 2001-2002 school year. The histogram on the left displays the ratios for the 24 states west of the Mississippi River, and the histogram on the right displays the ratios for the 26 states east of the Mississippi River.

a. Estimate the median P-T ratio for each group.

The question asks you to estimate the median, not compute it. For states west of the Mississippi (n = 24), the median falls between the 12th and 13th value in the ordered list, and both fall in the interval 15-16. For states east of the Mississippi (n = 26), the median falls between the 13th and 14th value, and both also fall in the interval 15-16. So both groups have a median of about 15 to 16 students per teacher.

b. Write a few sentences comparing the distributions of P-T ratios for states in the two groups (west and east) during the 2001-2002 school year.

Compare shape, center, and variability one at a time. Start with shape. The two histograms look different: West is unimodal and skewed to the right, while East is unimodal and nearly symmetric.

For center, part (a) showed the medians are about the same, between 15 and 16 for both groups.

For variability, look at how spread out the values are around the center. West values vary more than East values. Even though the data are grouped, you can still approximate the range. The range for West is at most 22 - 12 = 10, and the range for East is at most 19 - 12 = 7. So East has less variability than West.

c. Using your answers in parts (a) and (b), explain how you think the mean P-T ratio will compare for the two groups (west and east).

The two histograms have different shapes. Since West is skewed to the right, its mean will be pulled toward the high values, so the mean will be greater than the median. For East, since it is fairly symmetric, the mean will be close to the median. Putting that together, the mean for the West group will probably be greater than the mean for the East group.

Comparing Groups with Box Plots: Practice AP-Style Problem

A team of psychologists studied visualization in basketball, where players visualize making a basket before shooting. They ran an experiment in which 20 basketball players with similar abilities were randomly assigned to two groups. The 10 players in group 1 received visualization training, and the 10 players in group 2 did not.

Each player stood 22 feet from the basket at the same location on the court. Each player attempted to make the basket until two consecutive baskets were made. Players who received visualization training were instructed to use visualization techniques before each attempt. The total number of attempts, including the last two, was recorded for each player.

The total number of attempts for each of the 20 players is summarized in the following box plots.

There are two groups, with 10 players randomly assigned to each. Group 1 received visualization training and group 2 did not. Compare the box plots feature by feature.

Both groups have the same minimum number of attempts, but every other measure is different. At the first quartile, group 1 reached two consecutive baskets in 3 attempts, while group 2 took 4 attempts.

Now look at the median. The median is much lower for group 1 than for group 2. Group 1 has an outlier, but it is still less than the maximum of group 2. Across all five number summary values, group 1 is lower than group 2, which suggests the training had an effect. Note that this is a description of the sample, not yet a generalization about all players.

To answer the question, it is enough to compare medians. Because the median number of attempts for players who received visualization training (4) is less than the median for players who did not (7), players who received training tended to need fewer attempts to make two consecutive baskets.

Common Misconceptions

• Describing instead of comparing. Listing features of Group A, then separately listing features of Group B, does not count as comparing. Connect them with words like "higher," "lower," or "about the same."
• Reporting only the center. Comparing medians or means alone is incomplete. You also need shape and variability, plus unusual features.
• Forgetting context and units. A comparison without the variable and its units (like P-T ratio or number of attempts) is not a full answer.
• Mixing up skew and the mean's direction. In a right-skewed distribution, the mean is pulled toward the larger values (to the right of the median), not toward the peak.
• Using nonresistant measures with outliers. The mean, range, and standard deviation get pulled by outliers. When data are skewed or have outliers, lean on the median and IQR.
• Assuming different boxes with the same min are basically equal. Two groups can share a minimum and still differ a lot in median, quartiles, and spread. Check every part of the five number summary.

Related AP Statistics Guides

• Unit 1 Overview: Exploring One-Variable Data
• 1.1 Introducing Statistics: What Can We Learn from Data?
• 1.3 Representing a Categorical Variable with Tables
• 1.8 Graphical Representations of Summary Statistics
• 1.4 Representing a Categorical Variable with Graphs
• 1.6 Describing the Distribution of a Quantitative Variable