When does the mean beat the median?

Predict before you reveal

The same two estimators. Opposite outcomes.

Teach it or explore it

A five-minute activity

1. Predict: read each set of five observations and choose mean or median before opening the answer, or try a blind shareable challenge.
2. Test: open the workbench. Use the two example buttons, then paste a set of your own 3–40 numbers.
3. Challenge: remove the value farthest from the original median. Which estimate changes more? Then use five invented readings and a target to make a prediction challenge of your own.
4. Prove both outcomes: open Target Lens and keep all five readings fixed. Use its three exact targets to make the median win, make the mean win, and tie them. Certify 48–52 with reading drift 1, then compare the exact continuous threshold 13/10 with 1.299999, the last selectable strict guarantee, and 1.3, the first tie. Expand the threshold witness to inspect each signed reading change and both equal errors; copy the certificate for a class discussion. Press the exact-proof example: with target 0–1 and drift 4, the simpler 2ε bound stalls, yet the exact factor ranges prove the median wins throughout. Its exact threshold is 16/3, between six-decimal settings 5.333333 and 5.333334. At 16/3, readings 128/3, 131/3, 166/3, 166/3, 239/3 tie at target 0.
5. Move one reading: open Flip Point. Keep four readings and the target fixed, then find all four reversals as the fifth reading moves. A tie can also leave the winner unchanged on both sides.
6. See the whole decision: open the Decision Field. Drag the target and fifth reading together. Which ties occur at just one target, and which fifth readings tie for every target?
7. Extend: open the outcome atlas. At fifth reading 50, why can the mean have lower average error even when the median wins more individual cases? Select a cell, then open it in the full lab.

The generated cohort is deterministic toy data, not a sample of real observers. Workbench analysis runs in your browser. A challenge link includes all five values and the target in its URL, so use public or invented values only.

For educators and writers

Let your audience make the prediction.

For a paper activity, the three-case worksheet and worked answer key pair each invented set with a blind online check. Download the two-page PDF for predictable classroom printing.

This compact card works inside a webpage or course page that accepts HTML. Try the sample below, then open the full challenge to choose an exact case and copy its embed code. The visitor's answer stays in their browser; the widget uses no account or analytics.

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