A boxplot is a picture of the five-number summary: minimum, Q₁, median, Q₃, and maximum. A cumulative relative frequency graph shows what percent of the data falls at or below each value, so you can read percentiles straight off it. Edit the data and watch both graphs rebuild.
The box runs from Q₁ to Q₃ with a line at the median. Whiskers reach out to the most extreme values that are not outliers.
Each step on this graph is one data value. The height at a value x is the proportion of the data at or below x. The value at height 0.25 is about Q₁, 0.50 is about the median, and 0.75 is about Q₃.
These five values split the data into four parts, each holding about 25% of the observations. The boxplot draws exactly these five numbers, plus any outliers.
Each whisker and each half of the box holds about a quarter of the data. A longer section means those values are more spread out, not that there are more of them.
On the AP exam, boxplots are usually modified: values beyond the fences are drawn as separate points, and each whisker stops at the most extreme value that is not an outlier.
If the right whisker and the right half of the box are longer, the distribution is likely skewed right. If the left side is longer, it is likely skewed left. Similar sides suggest rough symmetry.
A boxplot does not show the sample size, gaps, clusters, or multiple peaks. Two very different data sets can have identical boxplots. Use a dotplot or histogram to see the full shape.
The pth percentile is the value with p% of the data at or below it. On a cumulative relative frequency graph, go across from p% on the vertical axis, then down to the value.
Typical boxplot and percentile questions, with the work written out.
| Question | Work | Answer |
|---|---|---|
| Five-number summary: 10, 18, 25, 31, 70. Is 70 an outlier? | IQR = 31 − 18 = 13. High fence = 31 + 1.5(13) = 50.5 | Yes, 70 > 50.5 |
| Same data: where does the right whisker end? | At the largest value that is ≤ 50.5 | The largest non-outlier, not at 70 |
| What percent of values are between Q₁ and the max? | Q₁ to median 25%, median to Q₃ 25%, Q₃ to max 25% | About 75% |
| Right whisker and right half of the box are much longer than the left. | Upper values are more spread out | Skewed right |
| A cumulative graph is at 0.80 when x = 45 minutes. Interpret. | 80% of values are at or below 45 | 45 minutes is the 80th percentile |
| A cumulative graph reads 0.30 at x = 12 and 0.70 at x = 20. What percent are between 12 and 20? | 0.70 − 0.30 | About 40% |
| Can you find the mean from a boxplot? | A boxplot shows only the five-number summary | No, the mean is not shown |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 1.8. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 1 terminology guide.