Read the box before the whiskers
| Element | What it usually represents | What to check |
|---|---|---|
| Lower edge of box | First quartile, near the 25th percentile | The software’s quartile algorithm can differ slightly in small samples. |
| Line inside box | Median, or 50th percentile | Its position within the box helps show asymmetry. |
| Upper edge of box | Third quartile, near the 75th percentile | Together with the lower quartile it defines the IQR. |
| Box height | Interquartile range: Q3 − Q1 | The middle half of the observed values spans this distance. |
| Whiskers | Convention-dependent limits | Never infer their rule from appearance alone. |
| Separate points | Values beyond the plotted whisker rule | They are flagged observations, not automatically errors. |
Start with the median and IQR. If the median lies near the center of the box and the two halves of the box are similar, the middle half of the data is fairly balanced around the median. If one half is much longer, the central data are more spread in that direction. Then inspect the whiskers and any separately plotted points.
The whisker conventions are not interchangeable
| Whisker rule | What the whiskers reach | Source/context | Category |
|---|---|---|---|
| Tukey-style 1.5×IQR | Most extreme observed points still inside Q1 − 1.5×IQR and Q3 + 1.5×IQR | Tukey tradition; R documents this as its positive coef=1.5 default1 2 | RULE OF THUMB |
| Minimum and maximum | Smallest and largest observed values | R uses this behavior when coef=0; some products expose it as a named option2 | SOFTWARE DEFAULT |
| Percentile pair | Selected lower and upper percentiles | Matplotlib accepts percentile pairs through its whis option3 | SOFTWARE DEFAULT |
| Standard-deviation or standard-error range | Software-defined limits based on SD or SE rather than an IQR fence | Origin exposes SD and SE among its whisker-range options4 | SOFTWARE DEFAULT |
The fourth form no longer communicates the same quartile-and-IQR geometry. If the caption or software documentation does not define the whiskers, the safe statement is simply that the figure does not provide enough information to classify outside points.
Tukey’s original construction also differs from many modern summaries in two useful details. It used hinges, not every modern software definition of sample quartiles, and Tukey called the 1.5-step fence a “rule of thumb.” In his terminology, values could be “outside” or “far out”; the rule did not certify a universal class of erroneous observations. Tukey also distinguished a rule-governed schematic plot from free-form box-and-whisker plots that did not necessarily follow those rules.1 This is why 1.5×IQR is an attributed convention, not the definition of every box plot.
The 1.5×IQR rule flags; it does not diagnose
For Tukey-style whiskers, calculate the conceptual fences from the quartiles:
- lower fence: Q1 − 1.5×IQR;
- upper fence: Q3 + 1.5×IQR.
The plotted whisker normally stops at the most extreme observed value inside each fence, not at the numeric fence itself.2 Values outside are commonly drawn as individual points. Calling all such points “outliers” can imply more than the rule establishes. They may be valid tail observations, members of a subgroup, measurement problems, or observations that deserve a model-specific influence check. Do not remove them solely because a plotting convention separated them.
For model influence rather than distribution summaries, use residual plots and influence diagnostics. To see the distribution’s bins and modes directly, compare the box plot with a histogram.
Why two programs can draw different boxes
Even with identical data, figures can differ because software chooses different:
- quartile or hinge algorithms;
- whisker rules;
- handling of missing values;
- display of flagged observations;
- notch calculations;
- orientation and grouping rules.
R, for example, documents hinges that are close to—but can differ from—sample quartiles and notches extending approximately 1.58×IQR/√n around the median.2 A notch is an approximate visual aid, not a universal hypothesis test. If notches overlap or do not overlap, the exact method and sample size still matter.
Shape clues a box plot can and cannot provide
A long upper whisker, a median near the lower box edge, and several high-side flagged points can suggest right skew. The mirror pattern can suggest left skew. But a box plot compresses the data: two distributions with different modes or gaps can share the same quartiles and whiskers. Use a scatter plot when the relationship between two variables matters, and interpret spread on the original scale with standard deviation only when that summary fits the distribution.
A reliable reading sentence
A defensible description names the convention: “The median is 18, the middle 50% runs from 12 to 26, and three observations lie beyond the software’s 1.5×IQR whiskers.” It does not say: “Three values are bad and should be deleted.”