# Accuracy, trial count and uncertainty: methods and claim sheet

Published: 14 October 2026  
Evidence type: reproducible methods note with deterministic calculations  
Participant or customer data: none

## Calculation

For x correct responses among n eligible trials, the observed proportion is p = x/n. The downloadable calculator reports a two-sided Wilson score interval without continuity correction. It obtains the normal critical value from Python's standard-library NormalDist implementation; the default confidence level is 95%.

Run an example:

    python3 calculate.py 8 10

Expected output:

    observed=80.0%
    wilson_95%=49.0% to 94.3%

The `examples.csv` file contains the worked values shown in the paper. Values are rounded to one decimal percentage point for display; calculations use unrounded values.

## Interpretation boundary

The binomial calculation assumes eligible trials are independent and share one unchanged probability of a correct response. Adaptive difficulty, unequal items, learning, fatigue, serial dependence, changing time limits and differing omission rules can violate those assumptions.

An interval around one session is not, by itself, a test of improvement between sessions. It does not establish task validity, everyday transfer, a minimum useful number of trials or a training benefit.

## Reproducibility and review

- The calculator uses only the Python standard library.
- The CSV rows were checked against the calculator on 14 October 2026.
- No Altamens user records, analytics or game telemetry were used.
- Literature search, drafting, calculations and publication preparation were AI-assisted.
- There has been no independent human methods review or journal peer review.

## Primary references

1. Wilson EB. Probable Inference, the Law of Succession, and Statistical Inference. JASA. 1927;22(158):209–212. DOI: 10.1080/01621459.1927.10502953.
2. Newcombe RG. Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine. 1998;17(8):857–872. DOI: 10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E.
3. Brown LD, Cai TT, DasGupta A. Interval Estimation for a Binomial Proportion. Statistical Science. 2001;16(2):101–133. DOI: 10.1214/ss/1009213286.
4. Agresti A, Coull BA. Approximate Is Better than “Exact” for Interval Estimation of Binomial Proportions. The American Statistician. 1998;52(2):119–126. DOI: 10.1080/00031305.1998.10480550.