Build a box plot
Drag any dot left or right along the number line. Tap an empty spot above the line to add a dot, up to 15. The box plot on top redraws as you go, and the five numbers behind it are listed underneath. A dot that turns orange is an outlier.
Mission. Press Outlier. The dot at 19 is orange. Drag it slowly to the left. Predict the value where it stops being an outlier, then check. What happens to the median while you drag?
Answer
It turns blue at 14. Here Q1 = 7 and Q3 = 10, so IQR = 3 and the upper fence is 10 + 1.5 × 3 = 14.5. Anything above 14.5 is an outlier. The median stays at 9 the whole time: it only cares about the middle dot, so dragging the largest value around cannot move it.
Box: the middle half of the data. Orange line: the median. Dashed lines: the fences. Open orange circles: outliers.
What's going on
Sort the data. The median is the middle value, or halfway between the middle two. Now split the list into a lower half and an upper half, leaving the median out if the count is odd. The median of the lower half is the lower quartile Q1. The median of the upper half is the upper quartile Q3. With the smallest and largest values, that makes five numbers.
The box runs from Q1 to Q3, so it holds the middle half of the data. Its width is the interquartile range, IQR = Q3 − Q1. The whiskers reach out to the smallest and largest values that are not outliers.
The usual rule for an outlier: go 1.5 × IQR below Q1 and 1.5 × IQR above Q3. Those two marks are the fences. Any value outside them is flagged and drawn on its own.
Press Lopsided and look at the shape. The median sits near the left of the box and the right whisker is long. The plot is showing skew without you reading a single number.
Some books and calculators find quartiles a slightly different way. This demo uses the median-of-halves method above.