{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are three options for the null and corresponding alternative hypothesis that can be selected using the " }, { "__type": "ParamRef", "__tag": 4071, "name": "alternative" }, { "__type": "Text", "__tag": 4046, "value": " parameter." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "two-sided", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ": The null hypothesis is that the two distributions are identical, F(x)=G(x) for all x; the alternative is that they are not identical." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "less", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ": The null hypothesis is that F(x) >= G(x) for all x; the alternative is that F(x) < G(x) for at least one x." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "greater", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ": The null hypothesis is that F(x) <= G(x) for all x; the alternative is that F(x) > G(x) for at least one x." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that the alternative hypotheses describe the " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "CDFs" } ] }, { "__type": "Text", "__tag": 4046, "value": " of the underlying distributions, not the observed values. 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See Notes for a description of the available null and alternative hypotheses." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/stats/_stats_py.py", "item_line": 7417, "item_type": "function", "aliases": [ "scipy.stats.ks_1samp" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Suppose we wish to test the null hypothesis that a sample is distributed\naccording to the standard normal.\nWe choose a confidence level of 95%; that is, we will reject the null\nhypothesis in favor of the alternative if the p-value is less than 0.05.\n\nWhen testing uniformly distributed data, we would expect the\nnull hypothesis to be rejected.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import stats\nrng = np.random.default_rng()\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.ks_1samp(stats.uniform.rvs(size=100, random_state=rng),\n stats.norm.cdf)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIndeed, the p-value is lower than our threshold of 0.05, so we reject the\nnull hypothesis in favor of the default \"two-sided\" alternative: the data\nare *not* distributed according to the standard normal.\n\nWhen testing random variates from the standard normal distribution, we\nexpect the data to be consistent with the null hypothesis most of the time.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x = stats.norm.rvs(size=100, random_state=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.ks_1samp(x, stats.norm.cdf)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nAs expected, the p-value of 0.92 is not below our threshold of 0.05, so\nwe cannot reject the null hypothesis.\n\nSuppose, however, that the random variates are distributed according to\na normal distribution that is shifted toward greater values. In this case,\nthe cumulative density function (CDF) of the underlying distribution tends\nto be *less* than the CDF of the standard normal. 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