Comparative box plots show several data distributions on the same number scale so you can compare groups quickly and fairly. Each box plot summarizes a data set using the minimum, first quartile, median, third quartile, and maximum, with possible outliers shown separately. This makes it easier to compare centers, spreads, skew, and unusual values without listing every data point.
They are useful in science, economics, sports, and any setting where groups must be compared using data.
When three box plots are stacked on one horizontal axis, differences in median show which group tends to have larger values. Differences in box length and whisker length show variability within each group. A longer right whisker or a median closer to the left side of the box suggests right skew, while the opposite suggests left skew.
Outliers help identify values that may be unusual, important, or worth checking for error.
Key Facts
- The five-number summary is minimum, Q1, median, Q3, maximum.
- IQR = Q3 - Q1 measures the spread of the middle 50% of the data.
- Range = maximum - minimum measures the full spread of the data, excluding separated outliers if shown that way.
- Outlier rule: values below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR) are often labeled outliers.
- A higher median line means that group has a higher typical value on the shared scale.
- A longer box or longer whiskers mean greater variability in that part of the distribution.
Vocabulary
- Median
- The median is the middle value of a data set when the values are ordered from least to greatest.
- Quartile
- A quartile is a value that divides ordered data into four parts with about 25% of the data in each part.
- Interquartile range
- The interquartile range is the distance from the first quartile to the third quartile and describes the spread of the middle half of the data.
- Skew
- Skew describes a distribution that stretches farther on one side than the other instead of being roughly balanced.
- Outlier
- An outlier is a data value that is unusually far from the rest of the values in a distribution.
Common Mistakes to Avoid
- Comparing box plots drawn on different scales is wrong because the visual lengths no longer represent the same numerical distances.
- Treating the whole box as the range is wrong because the box only runs from Q1 to Q3 and contains the middle 50% of the data.
- Assuming a larger median means every value in that group is larger is wrong because the distributions can overlap.
- Ignoring outliers is wrong because outliers can reveal unusual cases, measurement errors, or important features of the data.
Practice Questions
- 1 Group A has Q1 = 12, median = 18, and Q3 = 25. Find the IQR and interpret what it means.
- 2 Group B has Q1 = 40 and Q3 = 56. Use the 1.5(IQR) rule to find the lower and upper outlier fences.
- 3 Three groups have medians of 22, 27, and 25, but Group C has the longest box and whiskers. Explain which group has the greatest typical value and which group has the greatest variability.