{"abstract":"Unequal arm sizes give a standard error that is neither pooled nor Welch.","category":"Experiment statistics","checks":8,"contract":"a is control, b treatment. With sample variances (divisor n - 1), t = (mean_b - mean_a) / sqrt(va/na + vb/nb) and Welch-Satterthwaite df = (va/na + vb/nb)^2 / ((va/na)^2/(na-1) + (vb/nb)^2/(nb-1)). Fewer than two values in an arm or zero total variance -> None. Return [round(t, 6), round(df, 6)].","contract_signature":"a, b","evaluation_group":"w2-experiment-statistics-welch-t","failed_approach":"Adding the two standard errors instead of the variances overstates uncertainty.","family":"w2-experiment-statistics-welch-t-welch-standard-error","id":"FA-74401","implementations":{"attempt":{"sha256":"58e516253e88ce2083956442f8a42f67531097115a8cdbc53430a14c6200a9c8","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(a, b):\n    na, nb = len(a), len(b)\n    if na < 2 or nb < 2:\n        return None\n    ma, mb = sum(a) / na, sum(b) / nb\n    va = sum((x - ma) ** 2 for x in a) / (na - 1)\n    vb = sum((x - mb) ** 2 for x in b) / (nb - 1)\n    sa, sb = va / na, vb / nb\n    if sa + sb == 0:\n        return None\n    t = (mb - ma) / (math.sqrt(sa) + math.sqrt(sb))\n    df = (sa + sb) ** 2 / (sa ** 2 / (na - 1) + sb ** 2 / (nb - 1))\n    return [round(t, 6), round(df, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('metric sample 1', [[0, 18, 6, 6], [3, 7, 11, 17, 25, 22]], [1.290726, 7.219744]),\n  ('metric sample 2', [[12, 18, 10, 0, 10], [15, 23, 29, 2, 16]], [1.302114, 6.806789]),\n  ('metric sample 3', [[5, 3], [21, 5, 12, 23]], [2.622967, 3.321892])],\n [('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 4', [[6, 14, 12, 14, 7, 11, 3], [17, 29, 25, 22, 10]], [3.004687, 5.927117]),\n  ('metric sample 5', [[15, 1, 9, 13, 9], [25, 29, 28, 1, 29]], [2.199917, 5.520904]),\n  ('metric sample 6', [[13, 17, 16, 14, 0, 13], [10, 0, 6, 6, 0, 16, 5, 16]], [-1.428489, 10.999617])],\n [('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 11', [[6, 10], [16, 12, 20, 6, 18]], [2.007859, 4.050223]),\n  ('metric sample 12', [[7, 0], [27, 17, 26, 19, 20, 24, 1]], [3.236139, 3.199156]),\n  ('metric sample 13', [[7, 13, 5, 4, 12], [6, 24, 6, 27, 4, 8, 3, 17, 7, 17]], [1.122702, 12.999998])],\n [('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 16',\n   [[13, 7, 4, 9, 13, 12], [20, 21, 17, 13, 14, 25, 13, 11, 16, 14, 6]],\n   [2.66816, 13.725637]),\n  ('metric sample 18', [[18, 12, 1, 1], [12, 28, 6, 18, 8, 4, 15, 27, 24]], [1.497776, 6.245264]),\n  ('metric sample 19', [[17, 1, 8, 10, 2], [22, 17]], [3.102706, 3.799185])],\n [('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 21', [[0, 8, 15, 10], [28, 21, 11, 23, 28, 28]], [3.601238, 6.91205]),\n  ('metric sample 25', [[17, 3], [0, 10, 29, 29, 25, 3, 18]], [0.751104, 1.981932]),\n  ('metric sample 26', [[1, 11], [15, 23, 16]], [2.143769, 1.522005])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"b2313cd6d9ad985b0f7e78098091d8978106baf73462af32a36269b1c31cd15f","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(a, b):\n    na, nb = len(a), len(b)\n    if na < 2 or nb < 2:\n        return None\n    ma, mb = sum(a) / na, sum(b) / nb\n    va = sum((x - ma) ** 2 for x in a) / (na - 1)\n    vb = sum((x - mb) ** 2 for x in b) / (nb - 1)\n    sa, sb = va / na, vb / nb\n    if sa + sb == 0:\n        return None\n    t = (mb - ma) / math.sqrt((va + vb) / (na + nb))\n    df = (sa + sb) ** 2 / (sa ** 2 / (na - 1) + sb ** 2 / (nb - 1))\n    return [round(t, 6), round(df, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('metric sample 1', [[0, 18, 6, 6], [3, 7, 11, 17, 25, 22]], [1.290726, 7.219744]),\n  ('metric sample 2', [[12, 18, 10, 0, 10], [15, 23, 29, 2, 16]], [1.302114, 6.806789]),\n  ('metric sample 3', [[5, 3], [21, 5, 12, 23]], [2.622967, 3.321892])],\n [('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 4', [[6, 14, 12, 14, 7, 11, 3], [17, 29, 25, 22, 10]], [3.004687, 5.927117]),\n  ('metric sample 5', [[15, 1, 9, 13, 9], [25, 29, 28, 1, 29]], [2.199917, 5.520904]),\n  ('metric sample 6', [[13, 17, 16, 14, 0, 13], [10, 0, 6, 6, 0, 16, 5, 16]], [-1.428489, 10.999617])],\n [('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 11', [[6, 10], [16, 12, 20, 6, 18]], [2.007859, 4.050223]),\n  ('metric sample 12', [[7, 0], [27, 17, 26, 19, 20, 24, 1]], [3.236139, 3.199156]),\n  ('metric sample 13', [[7, 13, 5, 4, 12], [6, 24, 6, 27, 4, 8, 3, 17, 7, 17]], [1.122702, 12.999998])],\n [('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('equal arms give zero t', [[1, 2, 3], [1, 2, 3]], [0.0, 4.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 16',\n   [[13, 7, 4, 9, 13, 12], [20, 21, 17, 13, 14, 25, 13, 11, 16, 14, 6]],\n   [2.66816, 13.725637]),\n  ('metric sample 18', [[18, 12, 1, 1], [12, 28, 6, 18, 8, 4, 15, 27, 24]], [1.497776, 6.245264]),\n  ('metric sample 19', [[17, 1, 8, 10, 2], [22, 17]], [3.102706, 3.799185])],\n [('unequal sizes and variances', [[1, 2, 3, 4], [2, 4, 6, 8, 10, 12]], [2.713602, 6.594671]),\n  ('two-point arms', [[0, 2], [5, 9]], [2.683282, 1.470588]),\n  ('one arm constant', [[3, 3, 3], [1, 5, 9]], [0.866025, 2.0]),\n  ('negative effect', [[10, 12, 14, 16], [1, 3, 2]], [-7.778175, 4.075472]),\n  ('single observation arm', [[1], [2, 3]], None),\n  ('metric sample 21', [[0, 8, 15, 10], [28, 21, 11, 23, 28, 28]], [3.601238, 6.91205]),\n  ('metric sample 25', [[17, 3], [0, 10, 29, 29, 25, 3, 18]], [0.751104, 1.981932]),\n  ('metric sample 26', [[1, 11], [15, 23, 16]], [2.143769, 1.522005])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"A deterministic toy experiment-analysis model with a stipulated contract; results are rounded and are not a substitute for a validated statistics package. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-experiment-statistics-welch-t-welch-standard-error","generated_at":"2026-09-29T14:48:56.560627+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Online experiment readouts drive launch decisions; a silent formula slip flips conclusions.","root_cause":"The standard error is sqrt((va + vb) / (na + nb)).","sha256":"3a666e7350655b81ceb7050afdb354b4219d2ba663bca956bc3d0b7cc60e6803","title":"Welch unequal-variance t statistic: The t denominator mixes variances before dividing · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":39.095,"exit_code":1,"observations":[{"actual":[2.070848,6.594671],"check":"unequal sizes and variances","expected":[2.713602,6.594671],"passed":false},{"actual":[2.0,1.470588],"check":"two-point arms","expected":[2.683282,1.470588],"passed":false},{"actual":[0.866025,2.0],"check":"one arm constant","expected":[0.866025,2.0],"passed":true},{"actual":[0.0,4.0],"check":"equal arms give zero t","expected":[0.0,4.0],"passed":true},{"actual":[-5.887564,4.075472],"check":"negative effect","expected":[-7.778175,4.075472],"passed":false},{"actual":[0.913214,7.219744],"check":"metric sample 1","expected":[1.290726,7.219744],"passed":false},{"actual":[0.942638,6.806789],"check":"metric sample 2","expected":[1.302114,6.806789],"passed":false},{"actual":[2.175666,3.321892],"check":"metric sample 3","expected":[2.622967,3.321892],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"unequal sizes and variances\", \"actual\": [2.070848, 6.594671], \"expected\": [2.713602, 6.594671], \"passed\": false}, {\"check\": \"two-point arms\", \"actual\": [2.0, 1.470588], \"expected\": [2.683282, 1.470588], \"passed\": false}, {\"check\": \"one arm constant\", \"actual\": [0.866025, 2.0], \"expected\": [0.866025, 2.0], \"passed\": true}, {\"check\": \"equal arms give zero t\", \"actual\": [0.0, 4.0], \"expected\": [0.0, 4.0], \"passed\": true}, {\"check\": \"negative effect\", \"actual\": [-5.887564, 4.075472], \"expected\": [-7.778175, 4.075472], \"passed\": false}, {\"check\": \"metric sample 1\", \"actual\": [0.913214, 7.219744], \"expected\": [1.290726, 7.219744], \"passed\": false}, {\"check\": \"metric sample 2\", \"actual\": [0.942638, 6.806789], \"expected\": [1.302114, 6.806789], \"passed\": false}, {\"check\": \"metric sample 3\", \"actual\": [2.175666, 3.321892], \"expected\": [2.622967, 3.321892], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":37.716,"exit_code":1,"observations":[{"actual":[3.59521,6.594671],"check":"unequal sizes and variances","expected":[2.713602,6.594671],"passed":false},{"actual":[3.794733,1.470588],"check":"two-point arms","expected":[2.683282,1.470588],"passed":false},{"actual":[1.224745,2.0],"check":"one arm constant","expected":[0.866025,2.0],"passed":false},{"actual":[0.0,4.0],"check":"equal arms give zero t","expected":[0.0,4.0],"passed":true},{"actual":[-10.510864,4.075472],"check":"negative effect","expected":[-7.778175,4.075472],"passed":false},{"actual":[1.837959,7.219744],"check":"metric sample 1","expected":[1.290726,7.219744],"passed":false},{"actual":[1.841468,6.806789],"check":"metric sample 2","expected":[1.302114,6.806789],"passed":false},{"actual":[3.257033,3.321892],"check":"metric sample 3","expected":[2.622967,3.321892],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"unequal sizes and variances\", \"actual\": [3.59521, 6.594671], \"expected\": [2.713602, 6.594671], \"passed\": false}, {\"check\": \"two-point arms\", \"actual\": [3.794733, 1.470588], \"expected\": [2.683282, 1.470588], \"passed\": false}, {\"check\": \"one arm constant\", \"actual\": [1.224745, 2.0], \"expected\": [0.866025, 2.0], \"passed\": false}, {\"check\": \"equal arms give zero t\", \"actual\": [0.0, 4.0], \"expected\": [0.0, 4.0], \"passed\": true}, {\"check\": \"negative effect\", \"actual\": [-10.510864, 4.075472], \"expected\": [-7.778175, 4.075472], \"passed\": false}, {\"check\": \"metric sample 1\", \"actual\": [1.837959, 7.219744], \"expected\": [1.290726, 7.219744], \"passed\": false}, {\"check\": \"metric sample 2\", \"actual\": [1.841468, 6.806789], \"expected\": [1.302114, 6.806789], \"passed\": false}, {\"check\": \"metric sample 3\", \"actual\": [3.257033, 3.321892], \"expected\": [2.622967, 3.321892], \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}