R Analyses.
One Click. Full Reproducibility.

import random, math, hashlib

def make(seed):
    rng = random.Random(seed)
    a = [round(rng.gauss(50, 10), 2) for _ in range(30)]
    b = [round(rng.gauss(55, 12), 2) for _ in range(30)]
    return a, b

def welch(a, b):
    na, nb = len(a), len(b)
    ma, mb = sum(a) / na, sum(b) / nb
    va = sum((x - ma) ** 2 for x in a) / (na - 1)
    vb = sum((x - mb) ** 2 for x in b) / (nb - 1)
    se2 = va / na + vb / nb
    t = (mb - ma) / math.sqrt(se2)
    df = se2 ** 2 / ((va / na) ** 2 / (na - 1) + (vb / nb) ** 2 / (nb - 1))
    return ma, mb, t, df

for run in (1, 2):
    a, b = make(42)
    ma, mb, t, df = welch(a, b)
    digest = hashlib.sha256(repr((a, b)).encode()).hexdigest()[:12]
    print(f"run {run}: mean A={ma:.2f} mean B={mb:.2f} diff={mb-ma:.2f} t={t:.3f} df={df:.1f} data={digest}")

Output when run with python3 -I:

run 1: mean A=50.23 mean B=55.05 diff=4.82 t=1.896 df=46.2 data=c8d11576df95
run 2: mean A=50.23 mean B=55.05 diff=4.82 t=1.896 df=46.2 data=c8d11576df95

Check the numbers: 55.05 minus 50.23 is 4.82. The data fingerprint (data=) is the same on both runs, so the inputs did not change. A report from MCP Analytics works the same way: the R code is included, so you can run it on your own machine and compare.

Updated November 2, 2026.

Sources: Student (1908), The Probable Error of a Mean; Welch (1947), The generalization of "Student's" problem when several different population variances are involved; Wasserstein and Lazar (2016), The ASA Statement on p-Values. These show who published each method and when.