{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The statistic is calculated as " }, { "__type": "InlineCode", "__tag": 4051, "value": "(np.mean(a) - popmean)/se" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineCode", "__tag": 4051, "value": "se" }, { "__type": "Text", "__tag": 4046, "value": " is the standard error. Therefore, the statistic will be positive when the sample mean is greater than the population mean and negative when the sample mean is less than the population mean." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Beginning in SciPy 1.9, " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.matrix" }, { "__type": "Text", "__tag": 4046, "value": " inputs (not recommended for new code) are converted to " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.ndarray" }, { "__type": "Text", "__tag": 4046, "value": " before the calculation is performed. In this case, the output will be a scalar or " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.ndarray" }, { "__type": "Text", "__tag": 4046, "value": " of appropriate shape rather than a 2D " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.matrix" }, { "__type": "Text", "__tag": 4046, "value": ". Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.ndarray" }, { "__type": "Text", "__tag": 4046, "value": " rather than a masked array with " }, { "__type": "InlineCode", "__tag": 4051, "value": "mask=False" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Array API Standard Support" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "ttest_1samp", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable " }, { "__type": "InlineCode", "__tag": 4051, "value": "SCIPY_ARRAY_API=1" }, { "__type": "Text", "__tag": 4046, "value": " and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported." } ] }, { "__type": "Code", "__tag": 4050, "value": "==================== ==================== ====================\nLibrary CPU GPU\n==================== ==================== ====================\nNumPy ✅ n/a \nCuPy n/a ✅ \nPyTorch ✅ ⛔ \nJAX ⚠️ no JIT ⚠️ no JIT \nDask ⚠️ computes graph n/a \n==================== ==================== ====================", "execution_status": null }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "See " }, { "__type": "InlineRole", "__tag": 4003, "value": "dev-arrayapi", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for more information." } ] } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "result", "annotation": "`~scipy.stats._result_classes.TtestResult`", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "An object with the following attributes:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "statistic" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "statistic" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "pvalue" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "pvalue" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "df" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "df" } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The object also has the following method:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "confidence_interval(confidence_level=0.95)" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Computes a confidence interval around the population mean for the given confidence level. The confidence interval is returned in a " }, { "__type": "InlineCode", "__tag": 4051, "value": "namedtuple" }, { "__type": "Text", "__tag": 4046, "value": " with fields " }, { "__type": "InlineRole", "__tag": 4003, "value": "low", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineRole", "__tag": 4003, "value": "high", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.10.0" } ] } ] } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Calculate the T-test for the mean of ONE group of scores." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "a", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Sample observations." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "popmean", "annotation": "float or array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Expected value in null hypothesis. If array_like, then its length along " }, { "__type": "ParamRef", "__tag": 4071, "name": "axis" }, { "__type": "Text", "__tag": 4046, "value": " must equal 1, and it must otherwise be broadcastable with " }, { "__type": "ParamRef", "__tag": 4071, "name": "a" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "axis", "annotation": "int or None, default: 0", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If " }, { "__type": "InlineCode", "__tag": 4051, "value": "None" }, { "__type": "Text", "__tag": 4046, "value": ", the input will be raveled before computing the statistic." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "nan_policy", "annotation": "{'propagate', 'omit', 'raise'}", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defines how to handle input NaNs." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "propagate" }, { "__type": "Text", "__tag": 4046, "value": ": if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "omit" }, { "__type": "Text", "__tag": 4046, "value": ": NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "raise" }, { "__type": "Text", "__tag": 4046, "value": ": if a NaN is present, a " }, { "__type": "InlineCode", "__tag": 4051, "value": "ValueError" }, { "__type": "Text", "__tag": 4046, "value": " will be raised." } ] } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "alternative", "annotation": "{'two-sided', 'less', 'greater'}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defines the alternative hypothesis. The following options are available (default is 'two-sided'):" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'two-sided': the mean of the underlying distribution of the sample is different than the given population mean (" }, { "__type": "ParamRef", "__tag": 4071, "name": "popmean" }, { "__type": "Text", "__tag": 4046, "value": ")" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'less': the mean of the underlying distribution of the sample is less than the given population mean (" }, { "__type": "ParamRef", "__tag": 4071, "name": "popmean" }, { "__type": "Text", "__tag": 4046, "value": ")" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "'greater': the mean of the underlying distribution of the sample is greater than the given population mean (" }, { "__type": "ParamRef", "__tag": 4071, "name": "popmean" }, { "__type": "Text", "__tag": 4046, "value": ")" } ] } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "keepdims", "annotation": "bool, default: False", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This is a test for the null hypothesis that the expected value (mean) of a sample of independent observations " }, { "__type": "ParamRef", "__tag": 4071, "name": "a" }, { "__type": "Text", "__tag": 4046, "value": " is equal to the given population mean, " }, { "__type": "ParamRef", "__tag": 4071, "name": "popmean" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "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": 6030, "item_type": "function", "aliases": [ "scipy.stats.ttest_1samp" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Suppose we wish to test the null hypothesis that the mean of a population\nis equal to 0.5. We choose a confidence level of 99%; that is, we will\nreject the null hypothesis in favor of the alternative if the p-value is\nless than 0.01.\n\nWhen testing random variates from the standard uniform distribution, which\nhas a mean of 0.5, we expect the data to be consistent with the null\nhypothesis most of the time.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import stats\nrng = np.random.default_rng()\nrvs = stats.uniform.rvs(size=50, random_state=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.ttest_1samp(rvs, popmean=0.5)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nAs expected, the p-value of 0.017 is not below our threshold of 0.01, so\nwe cannot reject the null hypothesis.\n\nWhen testing data from the standard *normal* distribution, which has a mean\nof 0, we would expect the null hypothesis to be rejected.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rvs = stats.norm.rvs(size=50, random_state=rng)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "stats.ttest_1samp(rvs, popmean=0.5)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIndeed, the p-value is lower than our threshold of 0.01, so we reject the\nnull hypothesis in favor of the default \"two-sided\" alternative: the mean\nof the population is *not* equal to 0.5.\n\nHowever, suppose we were to test the null hypothesis against the\none-sided alternative that the mean of the population is *greater* than\n0.5. Since the mean of the standard normal is less than 0.5, we would not\nexpect the null hypothesis to be rejected.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "stats.ttest_1samp(rvs, popmean=0.5, alternative='greater')\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nUnsurprisingly, with a p-value greater than our threshold, we would not\nreject the null hypothesis.\n\nNote that when working with a confidence level of 99%, a true null\nhypothesis will be rejected approximately 1% of the time.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rvs = stats.uniform.rvs(size=(100, 50), random_state=rng)\nres = stats.ttest_1samp(rvs, popmean=0.5, axis=1)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "np.sum(res.pvalue < 0.01)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIndeed, even though all 100 samples above were drawn from the standard\nuniform distribution, which *does* have a population mean of 0.5, we would\nmistakenly reject the null hypothesis for one of them.\n\n`ttest_1samp` can also compute a confidence interval around the population\nmean.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rvs = stats.norm.rvs(size=50, random_state=rng)\nres = stats.ttest_1samp(rvs, popmean=0)\nci = res.confidence_interval(confidence_level=0.95)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "ci\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThe bounds of the 95% confidence interval are the\nminimum and maximum values of the parameter `popmean` for which the\np-value of the test would be 0.05.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = stats.ttest_1samp(rvs, popmean=ci.low)\nnp.testing.assert_allclose(res.pvalue, 0.05)\nres = stats.ttest_1samp(rvs, popmean=ci.high)\nnp.testing.assert_allclose(res.pvalue, 0.05)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nUnder certain assumptions about the population from which a sample\nis drawn, the confidence interval with confidence level 95% is expected\nto contain the true population mean in 95% of sample replications.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "rvs = stats.norm.rvs(size=(50, 1000), loc=1, random_state=rng)\nres = stats.ttest_1samp(rvs, popmean=0)\nci = res.confidence_interval()\ncontains_pop_mean = (ci.low < 1) & (ci.high > 1)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "contains_pop_mean.sum()\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "a", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "popmean", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0" }, { "__type": "SigParam", "__tag": 4030, "name": "nan_policy", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "propagate" }, { "__type": "SigParam", "__tag": 4030, "name": "alternative", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "two-sided" }, { "__type": "SigParam", "__tag": 4030, "name": "keepdims", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "False" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "ttest_1samp" }, "references": null, "qa": "scipy.stats._stats_py:ttest_1samp", "arbitrary": [], "local_refs": [ "a", "alternative", "axis", "keepdims", "nan_policy", "popmean", "result" ] }