A quotient census of integer means
Why 20 Integer Scores Cannot Round to 35.04.Scores 0 through 70, ordinary rounding to two decimals
Most numbers a table can print were never available to be printed. This page draws the lattice of means a group of n integer scores can actually produce, then runs a complete published table through it. At n = 20, no set of scores from 0 through 70 has a mean that rounds to 35.04 to two decimals.
- The claim
- No sequence of 20 integer scores from 0 through 70 has an arithmetic mean that rounds to 35.04 to two decimals.
- Domain swept
- computing integer aggregate sums, S = 0, 1, ..., 1400.
- Method
- Every dataset maps to one integer sum. Enumerate every sum and round the exact quotient S / 20.
- Positive control
- The adjacent printed mean 35.05 (3.33). The search finds S = 701 and an integer-score SD witness. mean search: 1 found
- Planted census probe
- The ordinary aggregate-sum census searches S = 0..700 for the reachable target 35.05. Add the legitimate integer sum 701 to that candidate domain and the same census must find it. 0 found
- Result
- computing
- Bound
- The null is only for n = 20, integer scores from 0 through 70, and the stated rounding rule. Past those assumptions, this certificate says nothing.
The arithmetic is small enough to see. Add twenty integers and the sum is an integer. Divide by twenty and the mean advances in steps of one twentieth, or 0.05. A printer offers one hundred hundredth-place labels between 35 and 36. This group can land on twenty of them.
Draw the lattice
Inputs are limited to 1 ≤ n ≤ 200 and nonnegative means with at most two decimals. The displayed score bounds remain 0 through 70.
Show the complete reachable set from 0 through 70
Make the empty search succeed
An empty answer from code is not persuasive unless the code can also return a nonempty answer. The published control sits immediately beside the focal cell. The planted probe modifies only the aggregate sums searched by the ordinary census, not its acceptance test or result path.
Positive control: 35.05 (3.33)
First the sum search must find 701 / 20 = 35.05. Then the joint search must find 20 integers from 29 through 70 whose sample SD also rounds to 3.33.
Planted census probe: removable
The candidate domain starts with every integer sum from 0 through 700, so none prints as 35.05. Plant S = 701. The unmodified census must discover exactly one match through its ordinary loop. Remove the sum and the result must return to zero.
A complete table, not a corpus
The worked application is Table 1 in Liu et al. (2023). The article labels each group n = 20. Its explicit response-summing sentence supports a strict 24-cell domain: PSS, SAI, TAI, BDI, PA and NA. The live conditional 28-cell view also includes four SLSI cells because the article calls SLSI a total score, but it does not explicitly say that SLSI responses were summed. Accuracy and reaction time are visibly excluded because they are not integer totals.
| measure | group | test | printed mean | printed SD | integer sums swept | mean result |
|---|---|---|---|---|---|---|
| Computing the conditional 28-cell mean census. | ||||||
Excluded, 8 cells: ACC and RT, four cells each. Source-explicit domain, 24 cells: six summed-score measures, four cells each. Conditional extension, 4 cells: SLSI, marked with an asterisk, if its reported total is a raw item sum. The 28 included cells were transcribed; the 8 exclusions were reviewed and named, not transcribed into the live data. The displayed SD is context only and is not tested. The table transcription is adapted under CC BY 4.0 from Liu et al. (2023), Table 1. Bounds are named free choices used to make each sum census finite; changing them does not change reachability when the printed mean remains inside the bounds.
The emptiness ends at 23
A denominator change is the strongest mundane escape route because it preserves integer scores. Move n upward and the mean first becomes reachable at 23. Now add the printed sample SD, 5.06. The deeper search does not merely test the mean. It enumerates sorted integer-score states from 29 through 70 by participant count, sum and sum of squares, pruning only states that cannot reach the exact final moments.
Joint mean and SD frontier
No n from 1 through 22 reaches the mean. At 23, the live dynamic program must construct scores that print as 35.04 (5.06).
This does not show that 23 observations were used. It shows exactly where one arithmetic obstruction stops. A variable-specific n = 23 would make both printed moments jointly possible, while the printed column header says 20.
the check
Recomputed here
Waiting for the live census.
Complete domains
The focal certificate checks all 1,401 sums from 0 through 1,400. The table mean census checks ... bounded sums across 24 source-explicit cells plus 4 conditional SLSI cells. The frontier checks every denominator 1 through 23, then every retained sorted state needed for the joint witness.
Free choices
- ordinary nearest rounding is the opening rule
- two printed decimals define the opening precision
- PSS is conservatively bounded 0 through 70
- other score bounds are printed in each census row
- sample SD uses denominator n - 1
What is uncertain
- raw observations were not retrieved
- the cause of the printed mismatch is unknown
- an unstated denominator or preprocessing step is possible
- whether the SLSI total is a raw item sum is not explicit
- the 28-cell census tests means only, not the 28 printed SDs
- the census does not audit the article's inferential results
- ACC and RT are outside the integer-total domain
Refusal
The instrument refuses noninteger or out-of-range group sizes, means with more than two decimals, means outside the score bounds, and unsupported score bounds. It does not silently round an input before testing it.
Sources and licence
Liu et al. (2023), Frontiers in Neuroscience 17:1182728, Table 1 and methods, retrieved 11 August 2026. Article and adapted transcription are CC BY 4.0. The general check follows the integer-mean reachability principle described by Brown and Heathers (2017).