The Five-Number Summary and Box Plots: Reading Quartiles, IQR, and Outliers
October 21, 2026
The five-number summary describes a dataset using five values: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. Together they show the center, the spread, and the skew of the data without assuming any particular distribution. A box plot (or box-and-whisker plot) draws them: a box from Q1 to Q3 with a line at the median, and whiskers extending to the smallest and largest typical values, with unusual points marked individually.
Take the twelve values 3, 5, 7, 8, 9, 10, 12, 13, 15, 18, 21, and 45. The median is the average of the 6th and 7th values: (10 + 12) ÷ 2 = 11. The lower half (3, 5, 7, 8, 9, 10) has a median of 7.5, which is Q1. The upper half (12, 13, 15, 18, 21, 45) has a median of 16.5, which is Q3.
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1. Sort the data. 2. Find the median: the middle value, or the average of the two middle values for an even count. 3. Split the data into lower and upper halves (excluding the median if the count is odd, in the common method). 4. Q1 is the median of the lower half; Q3 is the median of the upper half. 5. The minimum and maximum are the smallest and largest values. Software packages may use different quartile definitions, giving slightly different Q1 and Q3 for small samples. The interquartile range is IQR = Q3 − Q1 = 16.5 − 7.5 = 9.
The IQR contains the middle 50% of the data and is a robust measure of spread, unlike the range or standard deviation, which are affected by extreme values.
| Statistic | Value | Meaning |
|---|---|---|
| Minimum | 3 | Smallest value |
| Q1 | 7.5 | 25% of values are below |
| Median | 11 | Middle value |
| Q3 | 16.5 | 75% of values are below |
| Maximum | 45 | Largest value |
| IQR | 9 | Spread of the middle half |
| Mean | 13.8 | Pulled up by the 45 |
Whiskers and Outliers
The standard (Tukey) box plot extends the whiskers to the most extreme data points within 1.5 × IQR from the box. Here the fences are Q1 − 1.5 × 9 = −6 and Q3 + 1.5 × 9 = 30. The lower whisker reaches the minimum, 3. The upper whisker stops at 21, the largest value below 30. The value 45 lies beyond the fence and is plotted as an individual point, flagged as a potential outlier. The mean of 13.8 exceeds the median of 11, a sign of right skew caused by the high value.
An outlier is not automatically an error. Investigate it: it may be a data entry mistake, or a genuine and important observation.
Reading Box Plots
Look at the position of the median line within the box: if it is near the center, the middle half is roughly symmetric; if it is closer to Q1, the data are right-skewed. Compare whisker lengths: a longer upper whisker suggests a longer upper tail. The width of the box shows spread. Box plots are compact enough to compare many groups side by side. For example, a box plot of salaries for several departments reveals differences in median pay, spread, and outliers at a glance, which means and bar charts hide.
Box plots do not show the number of observations or multimodality (several peaks). Overlay the individual points or use a histogram when those matter.
When to Use the Five-Number Summary
Use it for skewed data, data with outliers, and comparisons across groups. Report the median and IQR instead of the mean and standard deviation when the distribution is not symmetric, such as incomes, house prices, and waiting times. For symmetric data without outliers, the mean and standard deviation are efficient summaries. In a spreadsheet, use QUARTILE.INC, QUARTILE.EXC, or the BOXWHISKER chart type.
Frequently Asked Questions
What is the five-number summary?
The minimum, first quartile, median, third quartile, and maximum.
What is the interquartile range?
Q3 minus Q1, the spread of the middle half of the data.
How do I identify outliers with a box plot?
Values more than 1.5 times the IQR below Q1 or above Q3 are plotted as outliers.
Why do different programs give different quartiles?
There are several accepted methods for calculating quartiles, especially with small samples.
When is the median better than the mean?
When data are skewed or contain outliers.