AP Statistics Five-Number Summary, Boxplots and Comparing Distributions Lesson (Free PDF)
This boxplots compare lesson for AP Statistics covers five-number summary, boxplots and comparing distributions (Common Core AP STAT 1.8, AP STAT 1.9, HSS-ID.A.1, HSS-ID.A.2), with a free PDF to print.
A boxplot displays the five-number summary: minimum, Q1, median, Q3, and maximum. The box contains the middle half of the data, whiskers reach the most extreme non-outliers, and stars mark outliers found with the 1.5 IQR rule. To compare distributions, discuss center, spread, shape, and unusual features with comparison words and context.
AP Statistics · Common Core State Standards: AP STAT 1.8, AP STAT 1.9, HSS-ID.A.1, HSS-ID.A.2
Key facts in this boxplots compare lesson
- Five-number summary: minimum, Q1, median, Q3, maximum.
- Each quarter of the ordered data holds about 25% of the values.
- IQR = Q3 – Q1; outliers lie more than 1.5 IQR outside the box.
- Compare center, spread, shape, and unusual features using comparison words.
- Boxplots show skew and outliers but not the number of peaks.
What is the five-number summary?
The five-number summary lists the minimum, first quartile \(Q_1\), median, third quartile \(Q_3\), and maximum. Together they split the ordered data into four groups that each contain about 25% of the values.
The interquartile range is \(IQR=Q_3-Q_1=15.5-7=8.5\) minutes. The upper fence is \(15.5+1.5(8.5)=28.25\), so 30 is an outlier.
How do you draw a boxplot?
How do you compare distributions?
When you compare two groups, discuss center, spread, shape, and unusual features, and use comparison words such as “greater than” and “more variable than.” A list of two separate descriptions earns little credit.
- Center: the median of Class B (74) is a little higher than that of Class A (72).
- Spread: Class B has an IQR of 22 compared with 15 for Class A, so Class B is more variable.
- Shape: both are roughly symmetric, with the median near the middle of each box.
- Unusual features: neither plot shows outliers.
How do you read skew and outliers in a boxplot?
Look at the position of the median line inside the box and at the lengths of the whiskers. In the wait-time boxplot, the median 11 is closer to Q1 (distance 4) than to Q3 (distance 4.5), but the upper whisker and the outlier star stretch far to the right, so the distribution is skewed right. A star never ends a whisker: the whisker stops at the last value inside the fence, which is 18 here.
Worked example: for the wait times, Q1 = 7 and Q3 = 15.5, so the lower fence is \(7-12.75=-5.75\). No value is below it, which is why the lower whisker reaches the minimum, 3. Always compute both fences before you draw.
A boxplot hides the number of peaks. Use a histogram or dotplot when you need to see gaps or bimodality.
Boxplots compare lesson: questions and answers
What percent of the data lies inside the box?
The box runs from Q1 to Q3, so it contains about the middle 50% of the data. About 25% lies in each whisker region and the two halves of the box.
Where does a whisker end?
A whisker ends at the smallest or largest value that is not an outlier. Values beyond the 1.5 IQR fences are plotted separately as stars or dots.
How can I tell skewness from a boxplot?
If the median is closer to Q1 and the upper whisker is longer, the data are skewed right. If the median is closer to Q3 and the lower whisker is longer, the data are skewed left.
Why compare medians and IQRs instead of means?
The median and IQR are resistant, so they describe center and spread reliably when the data are skewed or contain outliers. The five-number summary uses exactly these statistics.
What should a comparison sentence look like?
Use a comparison word and the context, such as: The median wait at Clinic B (14 minutes) is higher than at Clinic A (11 minutes), and its times are more variable (IQR 9 compared with 6).
How to use this boxplots compare lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the boxplots compare lesson as a PDF with the button below and keep it with your notes.
Download the boxplots compare lesson as a PDF with the button below and keep it with your notes.
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AP Statistics Five-Number Summary, Boxplots and Comparing Distributions Lesson
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