The Havant projectA median in five numbers, and the order that breaks it
Almost a year of Hermitage Stream flows, an exact answer, and an estimate that never stores the data.
The learner requests every 15-minute flow reading for the Havant gauge on the Hermitage Stream from the Environment Agency's hydrology service, from midnight on 1 September 2025 to 09:15 on 15 August 2026. That is 33,446 rows, of which 33,443 hold a value. The Agency flags 29,121 as good and 4,321 as not yet checked; every unchecked reading dates from 1 July 2026 onward, so the learner notes that the newest figures may still change.
First the exact answer. Sorting all 33,443 readings puts the median flow at 0.098 cubic metres a second, while the mean is 0.347. The stream spends most of its time low and occasionally surges, peaking at 12.009 at 19:45 on 1 February 2026, and only 24.3% of readings lie above the mean. Then the learner codes the P-square algorithm. It holds five marker heights and their positions, and each new reading nudges the middle marker up or down using a curve fitted through its neighbours. Memory stays at ten numbers however long the stream runs.
P-square median against the exact median, same 33,443 readings, our Python run| Order the readings arrive in | P-square median | Exact median |
|---|
| Shuffled at random | 0.0976 | 0.098 |
| A hundred more shuffles | 0.0974 to 0.0991 | 0.098 |
| Sorted low to high | 0.137 | 0.098 |
| As recorded, September to August | 0.2969 | 0.098 |
| Sorted high to low | 0.3339 | 0.098 |
Shuffled, the estimate lands within two thousandths of the truth every time. In the real order it ends at 0.2969, three times too high. The monthly medians explain why: 0.022 in October 2025 but 1.461 in February 2026. The winter pushed the markers up, and the dry spring and summer brought them down only slowly, because P-square moves each marker by at most one position per reading. By the end of March its median stood at 0.4534. The method quietly assumes that any stretch of the stream looks like the whole, and a river with seasons does not.
The same happens further up. The true 90th percentile is 1.029; shuffled, P-square finds 1.0321, in time order 1.7675. For the 99th percentile the exact value is 2.775; shuffled 2.7574, in time order 2.2158. The learner ends by testing fixes, such as one estimator per month or a random buffer before the estimator, and measures what each costs in memory.
Ages 8 to 11
Line up number cards to find the middle, then try to track it while cards arrive one at a time.
Ages 11 to 15
Read the flow file in Python, sort it, and compare median and mean.
Ages 15 and up
Code P-square from the 1985 paper, feed it in four orders, and design a fix.
Data and method
Flow readings are from the Environment Agency Hydrology API under the Open Government Licence. The estimator follows Jain and Chlamtac, Communications of the ACM, 1985. The shuffles use seed 20260930, and every figure on this page comes from our own Python run.
