Direct answer
A box plot displays a median, the middle half of ordered observations, convention-based whiskers, and separately flagged values; it does not show every observation or automatically reveal distribution density.
Visual explanation
Raw observations compressed into a box plot
What this calculation tells you
Box plots provide a compact comparison of center, robust spread, asymmetry, and potential extremes. Their power is comparison, but compression hides clusters, gaps, and exact sample size unless additional marks are supplied.
Use a box plot to compare distributions on a common scale, then inspect raw points, violin/density shapes, or histograms when multimodality, gaps, and group size affect interpretation.
Where it is used
Manufacturing
Compare process measurements across machines or periods.
Research
Display robust group summaries before inferential analysis.
Education
Teach order statistics and information compression.
Operations
Compare skewed service or delivery times across teams.
When this guide helps
- Reading a published box-and-whisker chart.
- Comparing several groups on one scale.
- Investigating a separately plotted value.
- Choosing a companion plot that exposes hidden structure.
Start with the statistical question
Use a box plot to compare distributions on a common scale, then inspect raw points, violin/density shapes, or histograms when multimodality, gaps, and group size affect interpretation.
Each raw observation is connected to its role in the five-number summary. The figure then hides the points to show exactly which information remains and which information disappears.
Worked example
Datasets 1,2,3,4,20 and 1,2,3,4,5 can have similarly placed boxes near the lower values, but the first may flag 20. That flag asks for investigation; it does not authorize automatic deletion.
Assumptions that carry the result
The article uses the 1.5×IQR whisker convention. Other software may draw whiskers to extrema or defined percentiles, so captions and methods must be checked before comparing graphics.
Interpret the result without overreaching
Overlapping boxes do not establish no difference, and separated boxes do not prove a causal effect. A box plot is descriptive and does not replace a study-design-aware analysis.
- Assuming whiskers always equal minimum and maximum.
- Reading the space inside a box as uniform density.
- Comparing groups without showing sample sizes or raw data.
Worked case: skewed sample
Values are 1, 2, 3, 4, 5 and 100.
Median = 3.5. Under a median-of-halves convention Q 1 = 2 and Q 3 = 5, so IQR = 3; 100 lies far beyond Q 3+1.5IQR = 9.5.
The plot flags 100 under the 1.5-IQR rule.
Flagged does not mean erroneous or removable.
Reproduce this worked caseOpen Box Plot Calculator
Worked case: different data, similar box
Many datasets can share the same quartiles and whiskers.
A box plot omits exact frequencies between landmarks unless raw points are overlaid.
Matching boxes do not prove matching distributions.
Use sample size and raw/violin points when detail matters.
Reproduce this worked caseOpen Box Plot Calculator
box plots: compare assumptions, not just answers
State quartile and whisker rules. Compare groups with sample size, distribution shape and measurement context.
| Case | Calculation focus | Interpretation |
|---|---|---|
| 1, 2, 3, 4, 5, 100 | IQR 3 | 100 flagged |
| Another set | Same quartiles possible | Different internal values |
box plots: calculation checklist
- Quartile method named
- Whisker rule stated
- Sample size shown
- Flags not deleted automatically
- Raw distribution considered
Practical questions
Frequently asked questions
Are points beyond the whiskers errors?
No. They are observations beyond a convention-based fence and may be valid, important, or erroneous after investigation.
Can box plots show sample size?
Ordinary equal-width boxes do not. Add raw points, labels, or intentionally variable widths when sample size matters.
Do non-overlapping boxes prove significance?
No. Box separation is descriptive; inference depends on design, sample size, variability, and the chosen method.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.