The Five-Number Summary and Boxplots
Master the art of summarizing an entire dataset with just five numbers and visualizing distributions with boxplots.
The Five-Number Summary
Definition
The Five-Number Summary consists of five values that provide a complete snapshot of a dataset's center, spread, and range.
LogicLens: Why Five Numbers?
These five values give you a complete snapshot of any distribution:
- • Range: Max − Min tells you the total spread
- • Center: The Median shows where the middle is
- • Middle 50%: to captures the IQR
- • Quartile positions: Show how data is distributed relative to the center
Constructing a Modified Boxplot
Calculate the Five-Number Summary and IQR
Find Min, , Median, , Max, and calculate .
Determine the Fences
Draw the Box
Draw a box from to with a vertical line at the Median.
Draw the Whiskers
Critical: Whiskers extend to the smallest and largest values within the fences, NOT to the fence values themselves!
Mark Outliers
Any values outside the fences are marked as outliers, typically with an asterisk (*) or dot.
Boxplot Anatomy
Reading Shape from Boxplots
LogicLens: How to Read Skewness
The position of the median within the box and the relative lengths of the whiskers tell you about the distribution's shape.
Symmetric
- • Median is centered in the box
- • Whiskers are roughly equal
- • Both halves mirror each other
Skewed Right (Positive)
- • Median is left of center
- • Right whisker is longer
- • Tail stretches toward high values
Skewed Left (Negative)
- • Median is right of center
- • Left whisker is longer
- • Tail stretches toward low values
Try It Yourself
Interactive Boxplot Builder
Five-Number Summary
Boxplot Visualization
IQR & Fences
IQR: 82.8 - 59.5 = 23.3
Lower Fence: 59.5 - 1.5(23.3) = 24.6
Upper Fence: 82.8 + 1.5(23.3) = 117.6
Whiskers end at: 45 (low) and 95 (high)
Outliers
No outliers detected in this dataset.
Distribution Shape
The median is roughly centered in the box, and the whiskers are approximately equal in length.
Adaptive Assessment
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