Experiment 03 / Target Lens
Move the target.
Flip the answer.
Keep the same five readings. Shift only the reference point and watch the mean and median trade places. When they differ, there is an exact boundary between them.
Hold readings still / move one reference
Find the exact crossing
The colored regions show which estimate has less absolute error for each hypothetical target from 0 to 100. The rose line is your selected target.
- 48
- 49
- 50
- 51
- 85
Slide to explore; enter a value for a precise target. This is a hypothetical reference, not a measured truth.
Why the boundary is exact: each error is the distance from target to its estimate. Between two different estimates, those distances match at their midpoint: (mean + median) ÷ 2 = 53.3. On either side, the nearer estimate has smaller error. When mean and median are equal, every target ties.
Constructive proof / same five readings
Three targets. Three exact verdicts.
Place the target at either estimate or their midpoint to prove the possible outcomes.
Why these witnesses work for every fixed set of five readings:
|mean − target|² − |median − target|² = (mean − median)(mean + median − 2×target)
This proves a statement about the chosen readings and hypothetical targets. It does not verify a target or compare estimators on observed populations.
Whole-range certificate / uncertain target
What if the target is only known within a range?
Median throughoutSet the lowest and highest possible hypothetical targets. This certificate checks every real-valued target between them, including the endpoints.
The entire range stays below the crossover.
At 48: Median. At 52: Median.
For distinct estimates, the squared-error difference has one zero, at their midpoint; coincident estimates tie everywhere. Very narrow bands are widened on the ruler for visibility; the certificate uses exact six-decimal endpoints and assumes no probability distribution or verified target.
Reading uncertainty / universal guarantee
What if every reading can drift?
Median guaranteedAllow each of the five readings to change independently by up to the same amount. The exact certificate checks every allowed configuration and every real target in the range above.
Exact factor ranges / all allowed readings
The signs of both factors determine the verdict throughout the allowed region.
The reachable tie targets
A single target at zero driftSee the full target × drift map
The shaded sides guarantee the named winner for every allowed reading arrangement. Their mint boundary admits a tie, so the guarantee ends there. Gold begins where mean and median can coincide, making every target a possible tie. Rose marks your selected range.
Explore tie depth across every target
At the current drift, each row lets at most the named number of readings move. Mint marks targets where some allowed arrangement ties; rose marks your selected range. Gold means an arrangement with equal estimates can tie at any target. If a row is highlighted, it is the minimum budget for a tie in your range.
Show exact five-reading witness
The exact stability threshold
How far can every reading move before a tie is possible?
Two routes to a tie · the earlier one sets the threshold
Show exact readings and target
Show exact readings and target
Inspect the exact threshold witness
Includes both exact routes and rational tie witnesses. Check it offline with Python ↗
Independent rational cross-checkAudit both routes from a saved receipt ↗
Minimum support / exact subset proof
How many readings must move?
—Keep some readings fixed. For the drift and target range above, check every subset of the five readings, then construct a tie using the smallest possible number that may move.
Explore the reading–drift frontier
How much drift is needed if at most one, two, three, four, or all five readings may change? The gold fraction is the exact continuous boundary; the mint value is the first selectable six-decimal setting. Open a row to inspect its rational tie witness.
Inspect exact witness
Inspect exact witness
Inspect exact witness
Inspect exact witness
Inspect exact witness
Which reading can tip the result by itself?
Hold four readings fixed and allow only the named reading to move. Compare the exact continuous threshold, the first selectable six-decimal setting, and a rational tie witness for each one.
Inspect exact witness
Inspect exact witness
Inspect exact witness
Inspect exact witness
Inspect exact witness
Inspect the simpler 2ε bound Shortcut certifies Median
A positive lead remains throughout the target range.
|mean − target|² − |median − target|² = (mean − median)(mean + median − 2×target)
The sum and median are monotone in every reading, giving exact bounds for the second factor. The first factor is optimized over all 30 choices of a median and two lower readings; within each ordering, its extrema occur at the limiting median values. The subset proof runs this exact certificate for all 32 choices of movable readings, freezing the others. If a factor crosses zero, the displayed rational fraction locates a tie along a path between its extrema. These are hypothetical ranges and stipulated reading limits, not measured uncertainty or a verified target.
Three ways to test the rule
Choose a fixed set
For a prediction before the arithmetic, use the three-case student sheet. To hold the target still and move one reading instead, try Flip Point. To move both together, open the Decision Field.
Carry the experiment
Share an exact view
Keep these five readings, selected hypothetical target, target range, and allowed reading drift together in a link. A recipient can check every number.
The link contains all five readings, the selected target, both range bounds, and the reading drift. The JSON receipt includes exact factor bounds and a rational tie witness when applicable. Check a receipt in your browser, use the offline Node or Python checkers, or independently audit both drift thresholds from the saved receipt. Use invented or public data. These references are chosen what-if values, not measured truth.