Read the Marks Before Comparing the Spread
In Five-Number Summary and Boxplot Construction, you learned that a boxplot represents a distribution using its five-number summary. In Applying the 1.5 IQR Rule for Outliers, you used the quartiles and IQR to identify possible outliers. Now you will read those features directly from a boxplot and explain what the four sections between the minimum and maximum tell you.
A boxplot uses a numerical scale. Read the scale first, then locate the positions of the whisker endpoints, the two ends of the box, and the median line inside the box. These positions represent data values; the box and whiskers do not show every individual observation.
Find the Five Values and Two Measures of Spread
Start by matching each visual feature to its value. On a horizontal boxplot, values usually increase from left to right. On a vertical boxplot, they usually increase from bottom to top. A scale might label every value, or it might label only selected tick marks; in that case, estimate a plotted position between ticks only as precisely as the scale allows.
The range describes the distance from the minimum to the maximum. It uses the two extreme values. The interquartile range, or IQR, describes the distance from \(Q_1\) to \(Q_3\), so it measures the spread of the middle half of the data. Both are expressed in the original units.
The range includes the entire span of the data, so it can be strongly affected by extreme observations. The IQR focuses on the distance across the box. Neither number describes every detail of the distribution, but each summarizes a different part of its spread.
For a basic boxplot, the whiskers extend to the minimum and maximum. A modified boxplot follows the 1.5 IQR rule from the previous tutorial: a possible outlier is plotted separately, and a whisker stops at the most extreme observation that is not flagged. In that case, a whisker endpoint is not necessarily the minimum or maximum. If separate outlier points are shown, include the farthest such points when identifying the full data range.
What the Four 25 Percent Sections Mean
The five-number summary divides the ordered observations into four rank sections: minimum to \(Q_1\), \(Q_1\) to the median, the median to \(Q_3\), and \(Q_3\) to the maximum. Each section contains about one-fourth of the observations. The box, from \(Q_1\) to \(Q_3\), covers the middle half.
These are statements about the number of observations in sections of the ordered data, not about equal distances on the number line. For example, the section from \(Q_1\) to the median might span 3 units, while the section from the median to \(Q_3\) spans 12 units. Both sections still represent about 25 percent of the observations. A longer section indicates more numerical spread in that part of the distribution, not more observations.
Quartile values can vary with the convention used to calculate them, as discussed in the earlier five-number-summary tutorial. Tied values can also appear at the same plotted position. So “25 percent” is a useful description of the groups by rank, not a promise that each visible interval contains exactly one-fourth of distinct numerical values.
Worked Example: Reading a Boxplot of Commute Times
Worked Example: Reading a Boxplot of Commute Times
A boxplot summarizes fictional students’ one-way commute times to school. Its scale is in minutes. The left whisker ends at 5, the box runs from 10 to 18, the median line is at 14, and the right whisker ends at 29. Read the five-number summary, find both measures of spread, and interpret the four sections.
Plan. In this basic boxplot, the whiskers reach the minimum and maximum. Read the box edges as \(Q_1\) and \(Q_3\), and the line inside the box as the median. Then subtract the appropriate endpoints to calculate range and IQR.
Do. The five-number summary is minimum \(=5\) minutes, \(Q_1=10\) minutes, median \(=14\) minutes, \(Q_3=18\) minutes, and maximum \(=29\) minutes. The range is:
The subtraction checks because \(5+24=29\). The IQR is:
This checks because \(10+8=18\). The IQR of 8 minutes is the distance across the middle half of the commute times; the range of 24 minutes is the distance from the shortest to the longest commute.
The four rank sections are 5 to 10 minutes, 10 to 14 minutes, 14 to 18 minutes, and 18 to 29 minutes. Each represents about one-fourth of the ordered observations. Their numerical lengths are \(10-5=5\), \(14-10=4\), \(18-14=4\), and \(29-18=11\) minutes. The upper section is the longest, but it still represents about one-fourth of the observations—not more students than the other sections.
Conclude. The middle 50 percent of these students’ commute times span 8 minutes, and the full data span is 24 minutes. The section from \(Q_3\) to the maximum has the greatest numerical spread.
Worked Example: Reading a Modified Boxplot
Worked Example: Reading a Modified Boxplot
A modified boxplot summarizes fictional daily measurements of rainfall, in millimeters. The lower whisker ends at 2, the box extends from 4 to 9, the median is 6, and the upper whisker ends at 13. A separate point appears at 25. Identify the quartiles and the full range, and explain why the upper whisker endpoint is not the maximum.
Plan. The separate point indicates an observation flagged by the 1.5 IQR rule. As in the previous tutorial, the modified boxplot’s whiskers end at the most extreme observations that are not flagged. Use the separate point when finding the full maximum and range.
Do. The box edges give \(Q_1=4\) millimeters and \(Q_3=9\) millimeters; the median is 6 millimeters. The lower whisker endpoint is the minimum here, at 2 millimeters, because no separate lower outlier is shown. The maximum is the separate point at 25 millimeters, not the upper whisker endpoint of 13.
The full range is:
Check: \(2+23=25\). The IQR is:
Check: \(4+5=9\). The four quartile sections, by rank, run from 2 to 4, 4 to 6, 6 to 9, and 9 to 25 millimeters. Each represents about one-fourth of the observations. The final section’s numerical span is \(25-9=16\) millimeters, even though the upper whisker stops at 13; the flagged value at 25 is still part of the data.
Conclude. The middle 50 percent of the rainfall measurements span 5 millimeters, while the full range is 23 millimeters. The upper whisker ends at 13 because it reaches the greatest non-outlier; the separate observation at 25 is the maximum and must be included in the range.
Worked Example: Comparing Two Boxplots
Worked Example: Comparing Two Boxplots
Two fictional groups of plants are measured for height in centimeters. Group A’s boxplot has a minimum of 12, \(Q_1=16\), median 20, \(Q_3=24\), and maximum 30. Group B’s boxplot has a minimum of 10, \(Q_1=18\), median 21, \(Q_3=23\), and maximum 32. Compare their centers and spreads using the median, IQR, and range.
Plan. Calculate each group’s IQR from its box endpoints and each range from its minimum and maximum. Compare medians to describe typical center, and compare the IQRs and ranges to describe spread. Keep the comparison in centimeters and in the context of plant heights.
Do. For Group A:
The arithmetic checks: \(16+8=24\), and \(12+18=30\). For Group B:
These calculations check because \(18+5=23\) and \(10+22=32\). Group B’s median is 21 cm, compared with 20 cm for Group A, so the medians are close, with Group B’s one centimeter higher. Group A has the larger IQR, 8 cm rather than 5 cm, so its middle half is more spread out. Group B has the larger range, 22 cm rather than 18 cm, because its full span is wider.
The four quartile sections in each boxplot each represent about one-fourth of that group’s observations. Their visual lengths do not make the groups’ sample sizes comparable; a boxplot summarizes positions and spread, not the number of observations.
Conclude. The groups have similar medians, but Group A has a wider middle half of plant heights, while Group B has the wider overall range. A careful comparison states which measure of spread is being compared rather than claiming that one group is simply “more spread out” in every sense.
Common Mistakes and AP Exam Tips
- Reading the wrong mark. The box edges are \(Q_1\) and \(Q_3\); the line inside the box is the median. Read each mark from the labeled numerical scale.
- Calling the box the full range. The box shows the middle 50 percent. The range uses the minimum and maximum, including separately plotted outliers when present.
- Assuming a modified whisker is an extreme value. A modified whisker ends at the most extreme non-outlier. Check for separate points before identifying the minimum or maximum.
- Confusing range and IQR. Range is maximum minus minimum; IQR is \(Q_3-Q_1\). Name the measure and use the corresponding endpoints.
- Thinking longer quartile sections contain more data. Each section represents about one-fourth of the observations. A longer section represents more numerical spread, not a greater share of the data.
- Comparing only by appearance. Use the scale and report values or differences in context. The same-looking box widths can represent different amounts if the plots use different scales.
- Leaving out units. State, for example, that an IQR is 8 minutes or a range is 23 millimeters—not just “8” or “23.”
For a full-credit interpretation, identify the relevant boxplot features, show any requested subtraction, include units, and describe what the result means for the variable. When discussing quartile sections, say that they represent about one-fourth of the observations each; do not infer that equal shares must occupy equal distances on the scale.
Check Your Understanding
Use each boxplot description to identify values or explain what the sections represent.
- A basic boxplot has a minimum of 7, \(Q_1=11\), median 15, \(Q_3=19\), and maximum 28. Find its range and IQR, including units if the variable is measured in hours.
- In a boxplot, the section from the median to \(Q_3\) is longer than the section from \(Q_1\) to the median. What does that say about numerical spread? Does it mean the longer section contains more observations?
- A modified boxplot’s upper whisker ends at 40, with a separate point at 55. If the lower whisker ends at the minimum value of 8, what is the maximum and full range?
- Explain what the box from \(Q_1\) to \(Q_3\) represents in terms of the observations.
- Two groups have medians of 32 and 35 units, with IQRs of 6 and 11 units. Which group has the more spread-out middle half, and by how much?