statistics.correlation misclassifies tiny nonconstant inputs as constant

[augusto-rehfeldt]

Bug report

Bug description:

Documented behaviour: The correlation() docstring states: "Return the Pearson's correlation coefficient for two inputs." It also demonstrates "correlation(x, x)" returning "1.0".

Expected: Return approximately 1.0 without raising an exception.

Actual: Raises StatisticsError: at least one of the inputs is constant.

import math
import statistics
from fractions import Fraction

a = 1e-200
x = [-a, a]
if not (1e-200 <= a <= 1e-170 and all(map(math.isfinite, x))
        and len(x) >= 2 and len(set(x)) > 1):
    print("REFUTATION REJECTED: invalid input")
else:
    q = list(map(Fraction, x))
    mean = sum(q) / len(q)
    variance = sum((t - mean) ** 2 for t in q)
    expected = float(variance / variance)  # Exact Pearson self-correlation.
    try:
        result = statistics.correlation(x, x)
        actual = ("result", result)
        broken = not math.isclose(result, expected, rel_tol=1e-12)
    except Exception as e:
        actual = ("exception", type(e).__name__, str(e))
        broken = True
    if broken:
        print("REFUTATION CONFIRMED:", (x, x), "actual =", actual,
              "expected =", expected)
    else:
        print("REFUTATION REJECTED: actual matches exact Pearson self-correlation")

Output on Python 3.14.6 (Windows-11-10.0.26220-SP0), standard library statistics:

REFUTATION CONFIRMED: ([-1e-200, 1e-200], [-1e-200, 1e-200]) actual = ('exception', 'StatisticsError', 'at least one of the inputs is constant') expected = 1.0

This report was found and written by an automated property-testing tool I run (bugforge). The reproducer above was executed and its output is pasted unedited; no person reviewed the report before it was filed. The search script is in https://github.com/augusto-rehfeldt/bugforge-results/tree/main/statistics-20261003-045701-c3

CPython versions tested on:

3.14

Operating systems tested on:

Windows

[ZeroIntensity]

See #158748 (comment).