Percentiles and a box plot

Parametric equations

This page reads the order of a one-dimensional sample. It does not draw a point cloud or a regression line. Sort the $n$ values

The percentile $P_p$ is linear interpolation at index $(p/100)(n-1)$ in that sorted list, counting from 0. That is the inclusive percentile: $P_0$ is the minimum and $P_{100}$ is the maximum. The box plot uses the same rule at

The interquartile range is $\mathrm{IQR}=Q_3-Q_1$. Whiskers stop at the outermost values still inside $[Q_1-k\cdot \mathrm{IQR},Q_3+k\cdot \mathrm{IQR}]$. Points outside those fences are marked as outliers.

Interaction

• Sample size n, Spread s, and Skew γ: reshape the sample and watch the box and the whiskers follow
• Whisker multiple k: move the fences and see which points fall outside
• Percentiles: show $P_{10}$ and $P_{90}$ as extra cuts beside the box
• Ranks: switch on the ordered places of the same values
• Drag a point to change its value. Delete removes the selected point when more than five values remain. Shift left and Shift right translate the sample. Pull in and Pull apart scale it about the median. Reset rebuilds it from the sliders

What to notice

• The median is the middle of the ordered sample. An extreme value usually moves it less than the mean. This figure does not plot the mean.
• The box width $\mathrm{IQR}$ is the span from $P_{25}$ to $P_{75}$ on this rule. A long tail can leave the box and the whiskers telling different stories.
• The page keeps one percentile rule. Read the ordered places and the shape. It does not compare interpolation rules from other software.

Related

• Scatter diagram, correlation, and the regression line
• An outlier’s effect on the regression line
• Data analysis

Further reading

• Box plot (Wikipedia)
• Percentile (Wikipedia)