AP Syllabus focus: 'Histograms, side-by-side boxplots, and other displays can compare independent samples on center, variability, clusters, gaps, outliers, and other features.'
When two groups of numerical data are shown graphically, the goal is to compare whole distributions, not isolated values, and to support claims with clear visual evidence.
Purpose of Comparing Quantitative Graphs
Comparing quantitative graphs means examining two or more distributions to describe how the groups differ. The focus is on overall patterns, not a few individual observations. Good comparisons use clear comparative language and refer to what the display shows.
Graphs can reveal whether one group tends to have larger values, whether one group is more variable, and whether unusual values appear in one group but not another.
Independent Samples
A comparison is usually made between independent samples.
Independent samples: Two samples in which the observations in one sample do not determine or pair with the observations in the other sample.
This matters because the goal is to compare separate groups, not matched observations. If the samples are independent, differences in the graphs represent differences between groups rather than a pairing structure.
Graphs Commonly Used
Histograms
Histograms are useful for comparing the overall pattern of two quantitative distributions. They can show:
differences in center, such as one group being shifted right
differences in variability, such as one group covering a wider interval
visible clusters, gaps, or possible outliers
For a fair comparison, histograms should use the same horizontal scale and comparable interval widths.
Otherwise, apparent differences may come from the display rather than the data.
Side-by-Side Boxplots
Side-by-side boxplots are compact displays for direct comparison.
They are especially useful for comparing:
the median as a measure of center
the spread of the middle half of the data
overall spread and possible outliers
Boxplots are efficient when several groups are compared at once. They do not show every detail of shape, so they are strongest for center, variability, and unusual values.
Other Displays
Other displays, such as dotplots, can also compare quantitative data. These are helpful when seeing individual observations matters. Whatever display is used, the same variable must be measured for each group and the graphs must allow a fair visual comparison.
What to Compare
Center
The center describes a typical value for a distribution. Ask which group tends to have larger or smaller values overall. In a histogram, this may appear as one distribution being farther right or left. In side-by-side boxplots, the medians often give the clearest comparison of center.
This labeled boxplot diagram highlights how a boxplot encodes the five-number summary (minimum, , median, , maximum) on a scaled number line. Seeing how the median and quartiles are positioned helps students justify comparisons of center and variability when reading side-by-side boxplots. Source
A good statement compares both groups directly. It is not enough to describe each graph separately without linking them.
Variability
Variability refers to how spread out the data are. A group with greater variability is less consistent, while a group with smaller variability is more tightly clustered. In graphs, variability may appear as:
a wider overall range
a wider box or longer whiskers in a boxplot
a broader distribution in a histogram
When comparing variability, use the same scale. A compressed axis can make a distribution look less variable than it really is.
Clusters, Gaps, Outliers, and Other Features
Graphs can reveal clusters, where observations are concentrated, and gaps, where few or no values occur. These features may show that one group has structure that another group does not.
Outliers are unusually high or low observations relative to the rest of a distribution. In comparisons, outliers matter because they may make one group appear more spread out or more unusual. If one graph shows outliers and the other does not, that is often important to report.
Other visible features may include differences in overall shape, such as one group being more concentrated in a narrow region while another has a longer tail. Any claimed feature should be supported by the graph.
Writing Strong Comparisons
A strong comparison directly connects the groups and describes how they differ.
Use comparative wording: higher center, greater variability, more pronounced cluster, fewer outliers.
Refer to the display: state what the histogram, boxplot, or other graph shows.
Keep the comparison in context by naming the groups and the variable.
Mention more than one feature when appropriate, especially center and variability.
Strong statistical writing identifies a feature, compares the groups on that feature, and supports the claim with visual evidence.
Common Errors to Avoid
Describing instead of comparing: writing one sentence about each graph without stating how they differ.
Ignoring scale: comparing graphs that do not use the same axis or interval widths.
Focusing on single values: the goal is to compare distributions, not just the largest or smallest observation.
Overstating conclusions: a graph shows differences in the samples, but it does not by itself explain why those differences exist.
Missing unusual features: clusters, gaps, and outliers can be as important as center and spread.
FAQ
If the histograms show counts, the group with the larger sample will often have taller bars even when the shapes are similar.
For a fairer comparison:
use relative frequency histograms, or
compare proportions in intervals instead of raw counts.
This helps you compare distributions rather than just comparing how many observations were collected.
Yes. A boxplot reduces a distribution to a small set of summary features, so it can hide important structure.
For example, two groups might have similar medians and quartiles but differ in:
modality
gaps
clusters
tail behavior
That is why histograms or dotplots can reveal differences that side-by-side boxplots do not show clearly.
A dotplot is often better when the samples are small or moderate in size and you want to see the actual data values.
A dotplot can make it easier to notice:
repeated values
exact gaps
small clusters
individual extreme observations
Histograms are better for broader patterns in larger data sets, while boxplots are better for quick summaries across several groups.
Then you should not compare bar heights directly without thinking about what the axis means.
A count graph shows the number of observations.
A relative frequency graph shows the proportion of observations.
If one graph uses counts and the other uses relative frequencies, first mentally convert both to the same kind of scale before comparing. Otherwise, you may confuse differences in sample size with differences in distribution.
Look for a clear visual difference, not a tiny one that may depend on the graph style.
A stronger claim is justified when one group shows:
a noticeably wider range
a wider box in a boxplot
a broader histogram
more dispersion around its center
If the difference is slight, use cautious language such as “appears slightly more variable” rather than making an absolute claim.
Practice Questions
Two side-by-side boxplots compare daily homework time for Group A and Group B. Group A has a median of 35 minutes, a smaller box, and no outliers. Group B has a median of 42 minutes, a larger box, and one high outlier.
State two differences between the distributions.
1 mark for stating that Group B has a higher center or higher median than Group A.
1 mark for stating that Group B has greater variability or that Group B has an outlier while Group A does not.
Two histograms on the same scale compare backpack weights for students at School X and School Y. School X is roughly unimodal, centered near 11 pounds, and most values fall between 8 and 14 pounds. School Y is centered near 12.5 pounds, has values from 6 to 24 pounds, has a long right tail, and has a gap from 19 to 21 pounds.
Write a complete comparison of the two distributions in context. Your answer should address center, variability, and at least one other feature.
1 mark for giving a direct comparison in context between School X and School Y.
1 mark for correctly stating that School Y has the higher center.
1 mark for correctly stating that School Y has greater variability or spread.
1 mark for identifying a shape difference, such as School Y having a longer right tail or being more right-skewed.
1 mark for identifying another relevant feature, such as the gap in School Y or the fact that School X is more concentrated.
