{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__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": "CrossRef", "__tag": 4002, "value": "kurtosis", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "kurtosis" }, "kind": "local" }, { "__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": "kurtosis", "annotation": "array", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The kurtosis of values along an axis, returning NaN where all values are equal." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute the kurtosis (Fisher or Pearson) of a dataset." } ] } ], "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", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Data for which the kurtosis is calculated." } ] } ] }, { "__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": "fisher", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If True, Fisher's definition is used (normal ==> 0.0). If False, Pearson's definition is used (normal ==> 3.0)." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "bias", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If False, then the calculations are corrected for statistical bias." } ] } ] }, { "__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": "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": "Kurtosis is the fourth central moment divided by the square of the variance. If Fisher's definition is used, then 3.0 is subtracted from the result to give 0.0 for a normal distribution." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If bias is False then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "InlineRole", "__tag": 4003, "value": "kurtosistest", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to see if result is close enough to normal." } ] } ], "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": 1356, "item_type": "function", "aliases": [ "scipy.stats.kurtosis" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "In Fisher's definition, the kurtosis of the normal distribution is zero.\nIn the following example, the kurtosis is close to zero, because it was\ncalculated from the dataset, not from the continuous distribution.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.stats import norm, kurtosis\ndata = norm.rvs(size=1000, random_state=3)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "kurtosis(data)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThe distribution with a higher kurtosis has a heavier tail.\nThe zero valued kurtosis of the normal distribution in Fisher's definition\ncan serve as a reference point.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import matplotlib.pyplot as plt\nimport scipy.stats as stats\nfrom scipy.stats import kurtosis\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "x = np.linspace(-5, 5, 100)\nax = plt.subplot()\ndistnames = ['laplace', 'norm', 'uniform']\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "for distname in distnames:\n if distname == 'uniform':\n dist = getattr(stats, distname)(loc=-2, scale=4)\n else:\n dist = getattr(stats, distname)\n data = dist.rvs(size=1000)\n kur = kurtosis(data, fisher=True)\n y = dist.pdf(x)\n ax.plot(x, y, label=\"{}, {}\".format(distname, round(kur, 3)))\n ax.legend()\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThe Laplace distribution has a heavier tail than the normal distribution.\nThe uniform distribution (which has negative kurtosis) has the thinnest\ntail." } ], "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": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0" }, { "__type": "SigParam", "__tag": 4030, "name": "fisher", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "True" }, { "__type": "SigParam", "__tag": 4030, "name": "bias", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "True" }, { "__type": "SigParam", "__tag": 4030, "name": "nan_policy", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "propagate" }, { "__type": "SigParam", "__tag": 4030, "name": "keepdims", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "False" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "kurtosis" }, "references": [ ".. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard", " Probability and Statistics Tables and Formulae. Chapman & Hall: New", " York. 2000." ], "qa": "scipy.stats._stats_py:kurtosis", "arbitrary": [], "local_refs": [ "a", "axis", "bias", "fisher", "keepdims", "kurtosis", "nan_policy" ] }