AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-8)
Unit 1 · Topic 1.8
1.8 Graphical Representations of Summary Statistics for One Quantitative Variable
A boxplot draws the five-number summary, so you can see center, spread and skew at a glance and spot outliers. You'll also learn how the mean and median compare in skewed distributions, which lets you infer shape from summary statistics alone.
Key terms
- five-number summary
- boxplot
- quartiles (Q1 and Q3)
- mean vs. median and skew
The five-number summary
The five-number summary is minimum, Q1, median, Q3, maximum. It splits the sorted data into four parts that each hold about 25% of the values.
The four parts can have very different widths. A wide quarter means those values are spread out, not that it holds more data.
Finding it with an even number of values
With an even n, split the sorted data exactly in half. For 3, 5, 6, 8, 9, 11, 14, 20: the median is the average of 8 and 9, which is 8.5. The lower half is 3, 5, 6, 8, so Q1 = (5 + 6)/2 = 5.5. The upper half is 9, 11, 14, 20, so Q3 = (11 + 14)/2 = 12.5. The five-number summary is 3, 5.5, 8.5, 12.5, 20.
Quartiles don't have to be data values. Software sometimes uses a slightly different rule and gets a slightly different Q1 or Q3; that's expected.
Drawing a boxplot
Draw a number line with the variable and units. Draw a box from Q1 to Q3 with a line at the median. The box holds the middle 50% of the data, and its length is the IQR.
Whiskers run from the box out to the smallest and largest values. If there are outliers (by the 1.5 × IQR rule), the whiskers stop at the most extreme values that aren't outliers, and each outlier gets its own dot or asterisk. This version is called a modified boxplot.
A boxplot hides detail. It can't show gaps, clusters or two peaks, and it doesn't show the sample size.
Reading shape from a boxplot
If the median sits in the middle of the box and the whiskers are about equal, the distribution is roughly symmetric. If the right whisker is long and the median is closer to Q1, it's skewed right. If the left whisker is long and the median is closer to Q3, it's skewed left.
Mean vs. median and shape
Extreme values in a long tail pull the mean toward them, while the median stays put. So:
- Roughly symmetric: mean and median are close.
- Skewed right: mean is usually greater than the median.
- Skewed left: mean is usually less than the median.
| Shape | Typical relationship |
|---|---|
| Roughly symmetric | Mean ≈ median |
| Skewed right | Mean > median |
| Skewed left | Mean < median |
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Build a modified boxplot
Use the commute times from 1.7: 5, 8, 10, 12, 12, 15, 18, 20, 22, 25, 48 (minutes). Q1 = 10, median = 15, Q3 = 22, and 48 is an outlier by the 1.5 × IQR rule. Describe the modified boxplot and the shape.
Show the solutionHide the solution
- Step 1: Box from 10 to 22, with a line at 15.
- Step 2: Left whisker from 10 down to the minimum, 5 (no low outliers).
- Step 3: Right whisker from 22 to 25, the largest value that isn't an outlier (the upper fence is 40).
- Step 4: Plot 48 as a separate point.
- Step 5: The median (15) is closer to Q1 than Q3, the right side is longer, and there's a high outlier. The mean (17.7) is greater than the median (15). All of this points to a right-skewed distribution.
Answer: Box from 10 to 22 with median line at 15, whiskers to 5 and 25, and a separate point at 48. The distribution of commute times is skewed right with a high outlier.
- Example 2Calculator allowed
Trap: longer whisker means more data?
A boxplot of 40 test scores has min 48, Q1 = 70, median 78, Q3 = 82, max 90. A student says more students scored between 48 and 70 than between 82 and 90 because that part is longer. Is that correct?
Show the solutionHide the solution
- Step 1: Each section of a boxplot holds about 25% of the data, no matter how long it is.
- Step 2: So about 10 of the 40 students are in each whisker section.
- Step 3: The longer left whisker means the lowest quarter of scores is more spread out, not that it holds more students.
Answer: No. About 25% of the students (about 10) fall in each whisker region. The longer left whisker means the low scores are more spread out, and the distribution is skewed left.
Common mistakes
- Thinking a longer section of a boxplot holds more data. Each quarter holds about 25%.
- Extending a whisker to an outlier. Whiskers stop at the most extreme value that isn't an outlier.
- Claiming a boxplot shows gaps, clusters or bimodality. It can't.
- Assuming mean > median always means skewed right. It usually does, but it's a guideline, not a rule.
On the exam
- Questions often give the mean and median and ask about likely shape, or give a boxplot and ask what percent of values lie in a region. Use the 25% sections.
- When drawing a boxplot by hand, mark outliers separately and include a labeled, scaled axis.
Connected topics
Videos
Check yourself
4 questions on 1.8 Graphical Representations of Summary Statistics for One Quantitative Variable. Pick an answer to see if you got it, and why.
A boxplot of the scores of 200 students on a quiz shows Q1 = 12, median = 15 and Q3 = 24. The box is much longer to the right of the median than to the left. Which statement is true?
For a sample of 500 home sale prices in a county, the mean is $412,000 and the median is $338,000. Which shape best fits these values?
A boxplot shows the number of minutes 40 people waited for a bus. Which of the following can be found from the boxplot?
| Texts sent | Number of students |
|---|---|
| 0 to less than 20 | 14 |
| 20 to less than 40 | 22 |
| 40 to less than 60 | 10 |
| 60 to less than 80 | 6 |
| 80 to less than 100 | 3 |
| 100 to less than 120 | 0 |
| 120 to less than 140 | 1 |
Invented data: number of texts sent in one day by 56 students, shown as a frequency table for a histogram
Which statement about the mean and median of these data is most likely true?
0 of 4 answered