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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code> --- 数学统计函数</a><ul>
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<li><a class="reference internal" href="#averages-and-measures-of-central-location">平均值以及对中心位置的评估</a></li>
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<section id="module-statistics">
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<span id="statistics-mathematical-statistics-functions"></span><h1><a class="reference internal" href="#module-statistics" title="statistics: Mathematical statistics functions"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code></a> --- 数学统计函数<a class="headerlink" href="#module-statistics" title="永久链接至标题">¶</a></h1>
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<p><span class="versionmodified added">3.4 新版功能.</span></p>
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<p><strong>源代码:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.8/Lib/statistics.py">Lib/statistics.py</a></p>
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<hr class="docutils" />
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<p>该模块提供了用于计算数字 (<a class="reference internal" href="numbers.html#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a>-valued) 数据的数理统计量的函数。</p>
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<p>此模块并不是诸如 <a class="reference external" href="https://numpy.org">NumPy</a> ， <a class="reference external" href="https://www.scipy.org/">SciPy</a> 等第三方库或者诸如 Minitab ， SAS ， Matlab 等针对专业统计学家的专有全功能统计软件包的竞品。此模块针对图形和科学计算器的水平。</p>
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<p>除非明确注释，这些函数支持 <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> ， <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> ， <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">Decimal</span></code></a> 和 <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> 。当前不支持同其他类型（是否在数字塔中）的行为。混合类型的集合也是未定义的，并且依赖于实现。如果你输入的数据由混合类型组成，你应该能够使用 <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate"><span class="pre">map()</span></code></a> 来确保一个一致的结果，比如： <code class="docutils literal notranslate"><span class="pre">map(float,</span> <span class="pre">input_data)</span></code> 。</p>
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<section id="averages-and-measures-of-central-location">
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<h2>平均值以及对中心位置的评估<a class="headerlink" href="#averages-and-measures-of-central-location" title="永久链接至标题">¶</a></h2>
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<p>这些函数用于计算一个总体或样本的平均值或者典型值。</p>
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<table class="docutils align-default">
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<colgroup>
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<col style="width: 27%" />
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<col style="width: 73%" />
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</colgroup>
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<tbody>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a></p></td>
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<td><p>数据的算术平均数（“平均数”）。</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate"><span class="pre">fmean()</span></code></a></p></td>
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<td><p>快速的，浮点算数平均数。</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.geometric_mean" title="statistics.geometric_mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">geometric_mean()</span></code></a></p></td>
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<td><p>数据的几何平均数</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.harmonic_mean" title="statistics.harmonic_mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">harmonic_mean()</span></code></a></p></td>
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<td><p>数据的调和均值</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate"><span class="pre">median()</span></code></a></p></td>
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<td><p>数据的中位数（中间值）</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_low()</span></code></a></p></td>
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<td><p>数据的低中位数</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_high()</span></code></a></p></td>
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<td><p>数据的高中位数</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code></a></p></td>
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<td><p>分组数据的中位数，即第50个百分点。</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate"><span class="pre">mode()</span></code></a></p></td>
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<td><p>离散的或标称的数据的单个众数（出现最多的值）。</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.multimode" title="statistics.multimode"><code class="xref py py-func docutils literal notranslate"><span class="pre">multimode()</span></code></a></p></td>
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<td><p>离散的或标称的数据的众数列表（出现最多的值）。</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.quantiles" title="statistics.quantiles"><code class="xref py py-func docutils literal notranslate"><span class="pre">quantiles()</span></code></a></p></td>
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<td><p>将数据以相等的概率分为多个间隔。</p></td>
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</tr>
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</tbody>
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</table>
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</section>
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<section id="measures-of-spread">
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<h2>对分散程度的评估<a class="headerlink" href="#measures-of-spread" title="永久链接至标题">¶</a></h2>
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<p>这些函数用于计算总体或样本与典型值或平均值的偏离程度。</p>
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<table class="docutils align-default">
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<colgroup>
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<col style="width: 34%" />
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<col style="width: 66%" />
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</colgroup>
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<tbody>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.pstdev" title="statistics.pstdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">pstdev()</span></code></a></p></td>
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<td><p>数据的总体标准差</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a></p></td>
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<td><p>数据的总体方差</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">stdev()</span></code></a></p></td>
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<td><p>数据的样本标准差</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a></p></td>
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<td><p>数据的样本方差</p></td>
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</tr>
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</tbody>
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</table>
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</section>
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<section id="function-details">
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<h2>函数细节<a class="headerlink" href="#function-details" title="永久链接至标题">¶</a></h2>
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<p>注释：这些函数不需要对提供给它们的数据进行排序。但是，为了方便阅读，大多数例子展示的是已排序的序列。</p>
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<dl class="function">
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<dt id="statistics.mean">
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<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">mean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mean" title="永久链接至目标">¶</a></dt>
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<dd><p>返回 <em>data</em> 的样本算术平均数，形式为序列或迭代器。</p>
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<p>算术平均数是数据之和与数据点个数的商。通常称作“平均数”，尽管它指示诸多数学平均数之一。它是数据的中心位置的度量。</p>
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<p>若 <em>data</em> 为空，将会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
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<p>一些用法示例：</p>
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<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
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<span class="go">2.8</span>
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<span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">5.75</span><span class="p">])</span>
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<span class="go">2.625</span>
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<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
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<span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">21</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)])</span>
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<span class="go">Fraction(13, 21)</span>
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|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
|
||
<span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">"0.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"0.75"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"0.625"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"0.375"</span><span class="p">)])</span>
|
||
<span class="go">Decimal('0.5625')</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">注解</p>
|
||
<p>均值非常受异常值的影响并且这不是中心位置的可靠估计：均值不一定是数据点的典型示例。如需要更可靠的的估计，请参考 <a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate"><span class="pre">median()</span></code></a> 和 <a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate"><span class="pre">mode()</span></code></a> 。</p>
|
||
<p>样本均值给出了一个无偏向的真实总体均值的估计，因此当平均抽取所有可能的样本， <code class="docutils literal notranslate"><span class="pre">mean(sample)</span></code> 收敛于整个总体的真实均值。如果 <em>data</em> 代表整个总体而不是样本，那么 <code class="docutils literal notranslate"><span class="pre">mean(data)</span></code> 等同于计算真实整体均值 μ 。</p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.fmean">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">fmean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.fmean" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>将 <em>data</em> 转换成浮点数并且计算算术平均数。</p>
|
||
<p>此函数的运行速度比 <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a> 函数快并且它总是返回一个 <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>。 <em>data</em> 可以为序列或可迭代对象。 如果输入数据集为空，则会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fmean</span><span class="p">([</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.25</span><span class="p">])</span>
|
||
<span class="go">4.25</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="versionadded">
|
||
<p><span class="versionmodified added">3.8 新版功能.</span></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.geometric_mean">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">geometric_mean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.geometric_mean" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>将 <em>data</em> 转换成浮点数并且计算几何平均数。</p>
|
||
<p>几何平均值使用值的乘积表示 <em>数据</em> 的中心趋势或典型值（与使用它们的总和的算术平均值相反）。</p>
|
||
<p>如果输入数据集为空、包含零或包含负值则将引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。 <em>data</em> 可以是序列或可迭代对象。</p>
|
||
<p>无需做出特殊努力即可获得准确的结果。（但是，将来或许会修改。）</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">round</span><span class="p">(</span><span class="n">geometric_mean</span><span class="p">([</span><span class="mi">54</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="mi">36</span><span class="p">]),</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="go">36.0</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="versionadded">
|
||
<p><span class="versionmodified added">3.8 新版功能.</span></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.harmonic_mean">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">harmonic_mean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.harmonic_mean" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回 <em>data</em> 调和均值，该参数可以是序列或包含实数值的可迭代对象。</p>
|
||
<p>调和均值,也叫次相反均值，所有数据的倒数的算术平均数 <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a> 的倒数。比如说，数据 <em>a</em> ， <em>b</em> ， <em>c</em> 的调和均值等于 <code class="docutils literal notranslate"><span class="pre">3/(1/a</span> <span class="pre">+</span> <span class="pre">1/b</span> <span class="pre">+</span> <span class="pre">1/c)</span></code> 。如果其中一个值为零，结果为零。</p>
|
||
<p>调和均值是一种均值类型，是数据中心位置的度量。它通常适合于求比率和比例的平均值，比如速率。</p>
|
||
<p>假设一辆车在 40 km/hr 的速度下行驶了 10 km ，然后又以 60 km/hr 的速度行驶了 10 km 。车辆的平均速率是多少？</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">harmonic_mean</span><span class="p">([</span><span class="mi">40</span><span class="p">,</span> <span class="mi">60</span><span class="p">])</span>
|
||
<span class="go">48.0</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>假设一名投资者在三家公司各购买了等价值的股票，以 2.5， 3 ， 10 的 P/E (价格/收益) 率。投资者投资组合的平均市盈率是多少？</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">harmonic_mean</span><span class="p">([</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span> <span class="c1"># For an equal investment portfolio.</span>
|
||
<span class="go">3.6</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>如果 <em>data</em> 为空或者 任何一个元素的值小于零，会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> 。</p>
|
||
<p>当前算法在输入中遇到零时会提前退出。这意味着不会测试后续输入的有效性。（此行为将来可能会更改。）</p>
|
||
<div class="versionadded">
|
||
<p><span class="versionmodified added">3.6 新版功能.</span></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>使用普通的“取中间两数平均值”方法返回数值数据的中位数（中间值）。 如果 <em>data</em> 为空，则将引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。 <em>data</em> 可以是序列或可迭代对象。</p>
|
||
<p>中位数是衡量中间位置的可靠方式，并且较少受到极端值的影响。 当数据点的总数为奇数时，将返回中间数据点：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>当数据点的总数为偶数时，中位数将通过对两个中间值求平均进行插值得出：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
|
||
<span class="go">4.0</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>这适用于当你的数据是离散的，并且你不介意中位数不是实际数据点的情况。</p>
|
||
<p>如果数据是有序的（支持排序操作）但不是数字（不支持加法），请考虑改用 <a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_low()</span></code></a> 或 <a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_high()</span></code></a>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median_low">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median_low</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_low" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回数值数据的低中位数。 如果 <em>data</em> 为空则将引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。 <em>data</em> 可以是序列或可迭代对象。</p>
|
||
<p>低中位数一定是数据集的成员。 当数据点总数为奇数时，将返回中间值。 当其为偶数时，将返回两个中间值中较小的那个。</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_low</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
<span class="gp">>>> </span><span class="n">median_low</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>当你的数据是离散的，并且你希望中位数是一个实际数据点而非插值结果时可以使用低中位数。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median_high">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median_high</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_high" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回数据的高中位数。 如果 <em>data</em> 为空则将引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。 <em>data</em> 可以是序列或可迭代对象。</p>
|
||
<p>高中位数一定是数据集的成员。 当数据点总数为奇数时，将返回中间值。 当其为偶数时，将返回两个中间值中较大的那个。</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_high</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
<span class="gp">>>> </span><span class="n">median_high</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
|
||
<span class="go">5</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>当你的数据是离散的，并且你希望中位数是一个实际数据点而非插值结果时可以使用高中位数。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median_grouped">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median_grouped</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">interval=1</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_grouped" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回分组的连续数据的中位数，根据第 50 个百分点的位置使用插值来计算。 如果 <em>data</em> 为空则将引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。 <em>data</em> 可以是序列或可迭代对象。</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">52</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">53</span><span class="p">,</span> <span class="mi">54</span><span class="p">])</span>
|
||
<span class="go">52.5</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>在下面的示例中，数据已经过舍入，这样每个值都代表数据分类的中间点，例如 1 是 0.5--1.5 分类的中间点，2 是 1.5--2.5 分类的中间点，3 是 2.5--3.5 的中间点等待。 根据给定的数据，中间值应落在 3.5--4.5 分类之内，并可使用插值法来进行估算：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
|
||
<span class="go">3.7</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>可选参数 <em>interval</em> 表示分类间隔，默认值为 1。 改变分类间隔自然会改变插件结果：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="n">interval</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="go">3.25</span>
|
||
<span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="n">interval</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="go">3.5</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>此函数不会检查数据点之间是否至少相隔 <em>interval</em> 的距离。</p>
|
||
<div class="impl-detail compound">
|
||
<p><strong>CPython implementation detail:</strong> 在某些情况下，<a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code></a> 可以会将数据点强制转换为浮点数。 此行为在未来有可能会发生改变。</p>
|
||
</div>
|
||
<div class="admonition seealso">
|
||
<p class="admonition-title">参见</p>
|
||
<ul class="simple">
|
||
<li><p>"Statistics for the Behavioral Sciences", Frederick J Gravetter and
|
||
Larry B Wallnau (8th Edition).</p></li>
|
||
<li><p>Gnome Gnumeric 电子表格中的 <a class="reference external" href="https://help.gnome.org/users/gnumeric/stable/gnumeric.html#gnumeric-function-SSMEDIAN">SSMEDIAN</a> 函数，包括 <a class="reference external" href="https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html">这篇讨论</a>。</p></li>
|
||
</ul>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.mode">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">mode</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mode" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>从离散或标称的 <em>data</em> 返回单个出现最多的数据点。 此众数（如果存在）是最典型的值，并可用来度量中心的位置。</p>
|
||
<p>如果存在具有相同频率的多个众数，则返回在 <em>data</em> 中遇到的第一个。 如果想要其中最小或最大的一个，请使用 <code class="docutils literal notranslate"><span class="pre">min(multimode(data))</span></code> 或 <code class="docutils literal notranslate"><span class="pre">max(multimode(data))</span></code>。 如果输入的 <em>data</em> 为空，则会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
<p><code class="docutils literal notranslate"><span class="pre">mode</span></code> 将假定是离散数据并返回一个单一的值。 这是通常的学校教学中标准的处理方式：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mode</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>此众数的独特之处在于它是这个包中唯一还可应用于标称（非数字）数据的统计信息：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mode</span><span class="p">([</span><span class="s2">"red"</span><span class="p">,</span> <span class="s2">"blue"</span><span class="p">,</span> <span class="s2">"blue"</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">,</span> <span class="s2">"green"</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">])</span>
|
||
<span class="go">'red'</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="versionchanged">
|
||
<p><span class="versionmodified changed">在 3.8 版更改: </span>现在会通过返回所遇到的第一个众数来处理多模数据集。 之前它会在遇到超过一个的众数时引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.multimode">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">multimode</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.multimode" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回最频繁出现的值的列表，并按它们在 <em>data</em> 中首次出现的位置排序。 如果存在多个众数则将返回一个以上的众数，或者如果 <em>data</em> 为空则将返回空列表：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">multimode</span><span class="p">(</span><span class="s1">'aabbbbccddddeeffffgg'</span><span class="p">)</span>
|
||
<span class="go">['b', 'd', 'f']</span>
|
||
<span class="gp">>>> </span><span class="n">multimode</span><span class="p">(</span><span class="s1">''</span><span class="p">)</span>
|
||
<span class="go">[]</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="versionadded">
|
||
<p><span class="versionmodified added">3.8 新版功能.</span></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.pstdev">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">pstdev</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">mu=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pstdev" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回总体标准差（总体方差的平方根）。 请参阅 <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> 了解参数和其他细节。</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">pstdev</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">4.75</span><span class="p">])</span>
|
||
<span class="go">0.986893273527251</span>
|
||
</pre></div>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.pvariance">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">pvariance</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">mu=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pvariance" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回非空序列或包含实数值的可迭代对象 <em>data</em> 的总体方差。 方差或称相对于均值的二阶距，是对数据变化幅度（延展度或分散度）的度量。 方差值较大表明数据的散布范围较大；方差值较小表明它紧密聚集于均值附近。</p>
|
||
<p>如果给出了可选的第二个参数 <em>mu</em>，它通常是 <em>data</em> 的均值。 它也可以被用来计算相对于一个非均值点的二阶距。 如果该参数省略或为 <code class="docutils literal notranslate"><span class="pre">None</span></code> (默认值)，则会自动进行算术均值的计算。</p>
|
||
<p>使用此函数可根据所有数值来计算方差。 要根据一个样本来估算方差，通常 <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> 函数是更好的选择。</p>
|
||
<p>如果 <em>data</em> 为空则会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
<p>示例：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">]</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="go">1.25</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>如果你已经计算过数据的平均值，你可以将其作为可选的第二个参数 <em>mu</em> 传入以避免重复计算：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mu</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">mu</span><span class="p">)</span>
|
||
<span class="go">1.25</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>同样也支持使用 Decimal 和 Fraction 值：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">"27.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"34.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"41.75"</span><span class="p">)])</span>
|
||
<span class="go">Decimal('24.815')</span>
|
||
|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
|
||
<span class="go">Fraction(13, 72)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">注解</p>
|
||
<p>当调用时附带完整的总体数据时，这将给出总体方差 σ²。 而当调用时只附带一个样本时，这将给出偏置样本方差 s²，也被称为带有 N 个自由度的方差。</p>
|
||
<p>如果你通过某种方式知道了真实的总体平均值 μ，则可以使用此函数来计算一个样本的方差，并将已知的总体平均值作为第二个参数。 假设数据点是总体的一个随机样本，则结果将为总体方差的无偏估计值。</p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.stdev">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">stdev</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">xbar=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.stdev" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回样本标准差（样本方差的平方根）。 请参阅 <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> 了解参数和其他细节。</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">stdev</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">4.75</span><span class="p">])</span>
|
||
<span class="go">1.0810874155219827</span>
|
||
</pre></div>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.variance">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">variance</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">xbar=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.variance" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回包含至少两个实数值的可迭代对象 <em>data</em> 的样本方差。 方差或称相对于均值的二阶矩，是对数据变化幅度（延展度或分散度）的度量。 方差值较大表明数据的散布范围较大；方差值较小表明它紧密聚集于均值附近。</p>
|
||
<p>如果给出了可选的第二个参数 <em>xbar</em>，它应当是 <em>data</em> 的均值。 如果该参数省略或为 <code class="docutils literal notranslate"><span class="pre">None</span></code> (默认值)，则会自动进行均值的计算。</p>
|
||
<p>当你的数据是总体数据的样本时请使用此函数。 要根据整个总体数据来计算方差，请参见 <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a>。</p>
|
||
<p>如果 <em>data</em> 包含的值少于两个则会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
<p>示例：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mf">2.75</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">]</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="go">1.3720238095238095</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>如果你已经计算过数据的平均值，你可以将其作为可选的第二个参数 <em>xbar</em> 传入以避免重复计算：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
|
||
<span class="go">1.3720238095238095</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>此函数不会试图检查你所传入的 <em>xbar</em> 是否为真实的平均值。 使用任意值作为 <em>xbar</em> 可能导致无效或不可能的结果。</p>
|
||
<p>同样也支持使用 Decimal 和 Fraction 值：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">"27.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"34.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"41.75"</span><span class="p">)])</span>
|
||
<span class="go">Decimal('31.01875')</span>
|
||
|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)])</span>
|
||
<span class="go">Fraction(67, 108)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">注解</p>
|
||
<p>这是附带贝塞尔校正的样本方差 s²，也称为具有 N-1 自由度的方差。 假设数据点具有代表性（即为独立且均匀的分布），则结果应当是对总体方差的无偏估计。</p>
|
||
<p>如果你通过某种方式知道了真实的总体平均值 μ 则应当调用 <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> 函数并将该值作为 <em>mu</em> 形参传入以得到一个样本的方差。</p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.quantiles">
|
||
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">quantiles</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">*</em>, <em class="sig-param">n=4</em>, <em class="sig-param">method='exclusive'</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.quantiles" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>将 <em>data</em> 分隔为具有相等概率的 <em>n</em> 个连续区间。 返回分隔这些区间的 <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">-</span> <span class="pre">1</span></code> 个分隔点的列表。</p>
|
||
<p>将 <em>n</em> 设为 4 以使用四分位（默认值）。 将 <em>n</em> 设为 10 以使用十分位。 将 <em>n</em> 设为 100 以使用百分位，即给出 99 个分隔点来将 <em>data</em> 分隔为 100 个大小相等的组。 如果 <em>n</em> 小于 1 则将引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
<p><em>data</em> 可以是包含样本数据的任意可迭代对象。 为了获得有意义的结果，<em>data</em> 中数据点的数量应当大于 <em>n</em>。 如果数据点的数量小于两个则将引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
<p>分隔点是通过对两个最接近的数据点进行线性插值得到的。 例如，如果一个分隔点落在两个样本值 <code class="docutils literal notranslate"><span class="pre">100</span></code> 和 <code class="docutils literal notranslate"><span class="pre">112</span></code> 之间距离三分之一的位置，则分隔点的取值将为 <code class="docutils literal notranslate"><span class="pre">104</span></code>。</p>
|
||
<p><em>method</em> 用于计算分位值，它会由于 <em>data</em> 是包含还是排除总体的最低和最高可能值而有所不同。</p>
|
||
<p>默认 <em>method</em> 是 “唯一的” 并且被用于在总体中数据采样这样可以有比样本中找到的更多的极端值。落在 <em>m</em> 个排序数据点的第 <em>i-th</em> 个以下的总体部分被计算为 <code class="docutils literal notranslate"><span class="pre">i</span> <span class="pre">/</span> <span class="pre">(m</span> <span class="pre">+</span> <span class="pre">1)</span></code> 。给定九个样本值，方法排序它们并且分配一下的百分位： 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90% 。</p>
|
||
<p>将 <em>method</em> 设为 "inclusive" 可用于描述总体数据或已明确知道包含有总体数据中最极端值的样本。 <em>data</em> 中的最小值会被作为第 0 个百分位而最大值会被作为第 100 个百分位。 总体数据里处于 <em>m</em> 个已排序数据点中 <em>第 i 个</em> 以下的部分会以 <code class="docutils literal notranslate"><span class="pre">(i</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">(m</span> <span class="pre">-</span> <span class="pre">1)</span></code> 来计算。 给定 11 个样本值，该方法会对它们进行排序并赋予以下百分位: 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%。</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="go"># Decile cut points for empirically sampled data</span>
|
||
<span class="gp">>>> </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mi">105</span><span class="p">,</span> <span class="mi">129</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">108</span><span class="p">,</span> <span class="mi">92</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span>
|
||
<span class="gp">... </span> <span class="mi">100</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">105</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">109</span><span class="p">,</span> <span class="mi">76</span><span class="p">,</span> <span class="mi">119</span><span class="p">,</span> <span class="mi">99</span><span class="p">,</span> <span class="mi">91</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">129</span><span class="p">,</span>
|
||
<span class="gp">... </span> <span class="mi">106</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">84</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">74</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">106</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span>
|
||
<span class="gp">... </span> <span class="mi">111</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">102</span><span class="p">,</span> <span class="mi">121</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">88</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">106</span><span class="p">,</span> <span class="mi">95</span><span class="p">,</span>
|
||
<span class="gp">... </span> <span class="mi">103</span><span class="p">,</span> <span class="mi">107</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">109</span><span class="p">,</span> <span class="mi">104</span><span class="p">]</span>
|
||
<span class="gp">>>> </span><span class="p">[</span><span class="nb">round</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">quantiles</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">10</span><span class="p">)]</span>
|
||
<span class="go">[81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="versionadded">
|
||
<p><span class="versionmodified added">3.8 新版功能.</span></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
</section>
|
||
<section id="exceptions">
|
||
<h2>异常<a class="headerlink" href="#exceptions" title="永久链接至标题">¶</a></h2>
|
||
<p>只定义了一个异常：</p>
|
||
<dl class="exception">
|
||
<dt id="statistics.StatisticsError">
|
||
<em class="property">exception </em><code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">StatisticsError</code><a class="headerlink" href="#statistics.StatisticsError" title="永久链接至目标">¶</a></dt>
|
||
<dd><p><a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> 的子类，表示统计相关的异常。</p>
|
||
</dd></dl>
|
||
|
||
</section>
|
||
<section id="normaldist-objects">
|
||
<h2><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 对象<a class="headerlink" href="#normaldist-objects" title="永久链接至标题">¶</a></h2>
|
||
<p><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 工具可用于创建和操纵 <a class="reference external" href="http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm">随机变量</a> 的正态分布。 这个类将数据度量值的平均值和标准差作为单一实体来处理。</p>
|
||
<p>正态分布的概念来自于 <a class="reference external" href="https://en.wikipedia.org/wiki/Central_limit_theorem">中央极限定理</a> 并且在统计学中有广泛的应用。</p>
|
||
<dl class="class">
|
||
<dt id="statistics.NormalDist">
|
||
<em class="property">class </em><code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">NormalDist</code><span class="sig-paren">(</span><em class="sig-param">mu=0.0</em>, <em class="sig-param">sigma=1.0</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>返回一个新的 <em>NormalDist</em> 对象，其中 <em>mu</em> 代表 <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">算术平均值</a> 而 <em>sigma</em> 代表 <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">标准差</a>。</p>
|
||
<p>若 <em>sigma</em> 为负数，将会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>。</p>
|
||
<dl class="attribute">
|
||
<dt id="statistics.NormalDist.mean">
|
||
<code class="sig-name descname">mean</code><a class="headerlink" href="#statistics.NormalDist.mean" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>一个只读特征属性，表示特定正态分布的 <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">算术平均值</a>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="attribute">
|
||
<dt id="statistics.NormalDist.median">
|
||
<code class="sig-name descname">median</code><a class="headerlink" href="#statistics.NormalDist.median" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>一个只读特征属性，表示特定正态分布的 <a class="reference external" href="https://en.wikipedia.org/wiki/Median">中位数</a>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="attribute">
|
||
<dt id="statistics.NormalDist.mode">
|
||
<code class="sig-name descname">mode</code><a class="headerlink" href="#statistics.NormalDist.mode" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>一个只读特征属性，表示特定正态分布的 <a class="reference external" href="https://en.wikipedia.org/wiki/Mode_(statistics)">众数</a>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="attribute">
|
||
<dt id="statistics.NormalDist.stdev">
|
||
<code class="sig-name descname">stdev</code><a class="headerlink" href="#statistics.NormalDist.stdev" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>一个只读特征属性，表示特定正态分布的 <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">标准差</a>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="attribute">
|
||
<dt id="statistics.NormalDist.variance">
|
||
<code class="sig-name descname">variance</code><a class="headerlink" href="#statistics.NormalDist.variance" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>一个只读特征属性，表示特定正态分布的 <a class="reference external" href="https://en.wikipedia.org/wiki/Variance">方差</a>。 等于标准差的平方。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="method">
|
||
<dt id="statistics.NormalDist.from_samples">
|
||
<em class="property">classmethod </em><code class="sig-name descname">from_samples</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.from_samples" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>传入使用 <a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate"><span class="pre">fmean()</span></code></a> 和 <a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">stdev()</span></code></a> 基于 <em>data</em> 估算出的 <em>mu</em> 和 <em>sigma</em> 形参创建一个正态分布实例。</p>
|
||
<p><em>data</em> 可以是任何 <a class="reference internal" href="../glossary.html#term-iterable"><span class="xref std std-term">iterable</span></a> 并且应当包含能被转换为 <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> 类型的值。 如果 <em>data</em> 不包含至少两个元素，则会引发 <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>，因为估算中心值至少需要一个点而估算分散度至少需要两个点。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="method">
|
||
<dt id="statistics.NormalDist.samples">
|
||
<code class="sig-name descname">samples</code><span class="sig-paren">(</span><em class="sig-param">n</em>, <em class="sig-param">*</em>, <em class="sig-param">seed=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.samples" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>对于给定的平均值和标准差生成 <em>n</em> 个随机样本。 返回一个由 <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> 值组成的 <a class="reference internal" href="stdtypes.html#list" title="list"><code class="xref py py-class docutils literal notranslate"><span class="pre">list</span></code></a>。</p>
|
||
<p>当给定 <em>seed</em> 时，创建一个新的底层随机数生成器实例。 这适用于创建可重现的结果，即使对于多线程上下文也有效。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="method">
|
||
<dt id="statistics.NormalDist.pdf">
|
||
<code class="sig-name descname">pdf</code><span class="sig-paren">(</span><em class="sig-param">x</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.pdf" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>使用 <a class="reference external" href="https://en.wikipedia.org/wiki/Probability_density_function">概率密度函数 (pdf)</a>，计算一个随机变量 <em>X</em> 趋向于给定值 <em>x</em> 的相对可能性。 在数学意义上，它是当 <em>dx</em> 趋向于零时比率 <code class="docutils literal notranslate"><span class="pre">P(x</span> <span class="pre"><=</span> <span class="pre">X</span> <span class="pre"><</span> <span class="pre">x+dx)</span> <span class="pre">/</span> <span class="pre">dx</span></code> 的极限。</p>
|
||
<p>相对可能性的计算方法是用一个狭窄区间内某个样本出现的概率除以区间的宽度（因此使用“密度”一词）。 由于可能性是相对于其他点的，它的值可以大于 <cite>1.0</cite>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="method">
|
||
<dt id="statistics.NormalDist.cdf">
|
||
<code class="sig-name descname">cdf</code><span class="sig-paren">(</span><em class="sig-param">x</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.cdf" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>使用 <a class="reference external" href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">累积分布函数 (cdf)</a>，计算一个随机变量 <em>X</em> 小于等于 <em>x</em> 的概率。 在数学上，它表示为 <code class="docutils literal notranslate"><span class="pre">P(X</span> <span class="pre"><=</span> <span class="pre">x)</span></code>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="method">
|
||
<dt id="statistics.NormalDist.inv_cdf">
|
||
<code class="sig-name descname">inv_cdf</code><span class="sig-paren">(</span><em class="sig-param">p</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.inv_cdf" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>计算反向累积分布函数，也称为 <a class="reference external" href="https://en.wikipedia.org/wiki/Quantile_function">分位数函数</a> 或 <a class="reference external" href="https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/">百分点</a> 函数。 在数学上，它表示为 <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">P(X</span> <span class="pre"><=</span> <span class="pre">x)</span> <span class="pre">=</span> <span class="pre">p</span></code>。</p>
|
||
<p>找出随机变量 <em>X</em> 的值 <em>x</em> 使得该变量小于等于该值的概率等于给定的概率 <em>p</em>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="method">
|
||
<dt id="statistics.NormalDist.overlap">
|
||
<code class="sig-name descname">overlap</code><span class="sig-paren">(</span><em class="sig-param">other</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.overlap" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>测量两个正态概率分布之间的一致性。 返回介于 0.0 和 1.0 之间的值，给出 <a class="reference external" href="https://www.rasch.org/rmt/rmt101r.htm">两个概率密度函数的重叠区域</a>。</p>
|
||
</dd></dl>
|
||
|
||
<dl class="method">
|
||
<dt id="statistics.NormalDist.quantiles">
|
||
<code class="sig-name descname">quantiles</code><span class="sig-paren">(</span><em class="sig-param">n=4</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.quantiles" title="永久链接至目标">¶</a></dt>
|
||
<dd><p>将指定正态分布划分为 <em>n</em> 个相等概率的连续分隔区。 返回这些分隔区对应的 (n - 1) 个分隔点的列表。</p>
|
||
<p>将 <em>n</em> 设为 4 以使用四分位（默认值）。 将 <em>n</em> 设为 10 以使用十分位。将 <em>n</em> 设为 100 以使用百分位，即给出 99 个分隔点来将正态分布分隔为 100 个大小相等的组。</p>
|
||
</dd></dl>
|
||
|
||
<p><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 的实例支持加上、减去、乘以或除以一个常量。 这些运算被用于转换和缩放。 例如：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">temperature_february</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span> <span class="c1"># Celsius</span>
|
||
<span class="gp">>>> </span><span class="n">temperature_february</span> <span class="o">*</span> <span class="p">(</span><span class="mi">9</span><span class="o">/</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">32</span> <span class="c1"># Fahrenheit</span>
|
||
<span class="go">NormalDist(mu=41.0, sigma=4.5)</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>不允许一个常量除以 <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 的实例，因为结果将不是正态分布。</p>
|
||
<p>由于正态分布是由独立变量的累加效应产生的，因此允许表示为 <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 实例的 <a class="reference external" href="https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables">两组独立正态分布的随机变量相加和相减</a>。 例如：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">birth_weights</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.1</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">,</span> <span class="mf">2.4</span><span class="p">,</span> <span class="mf">2.7</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">])</span>
|
||
<span class="gp">>>> </span><span class="n">drug_effects</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">combined</span> <span class="o">=</span> <span class="n">birth_weights</span> <span class="o">+</span> <span class="n">drug_effects</span>
|
||
<span class="gp">>>> </span><span class="nb">round</span><span class="p">(</span><span class="n">combined</span><span class="o">.</span><span class="n">mean</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="go">3.1</span>
|
||
<span class="gp">>>> </span><span class="nb">round</span><span class="p">(</span><span class="n">combined</span><span class="o">.</span><span class="n">stdev</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="go">0.5</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="versionadded">
|
||
<p><span class="versionmodified added">3.8 新版功能.</span></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<section id="normaldist-examples-and-recipes">
|
||
<h3><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 示例和用法<a class="headerlink" href="#normaldist-examples-and-recipes" title="永久链接至标题">¶</a></h3>
|
||
<p><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 适合用来解决经典概率问题。</p>
|
||
<p>举例来说，如果 <a class="reference external" href="https://nces.ed.gov/programs/digest/d17/tables/dt17_226.40.asp">SAT 考试的历史数据</a> 显示分数呈平均值为 1060 且标准差为 195 的正态分布，则可以确定考试分数处于 1100 和 1200 之间的学生的百分比舍入到最接近的整数应为：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">sat</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">1060</span><span class="p">,</span> <span class="mi">195</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">fraction</span> <span class="o">=</span> <span class="n">sat</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">1200</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">-</span> <span class="n">sat</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">1100</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="nb">round</span><span class="p">(</span><span class="n">fraction</span> <span class="o">*</span> <span class="mf">100.0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="go">18.4</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>求 SAT 分数的 <a class="reference external" href="https://en.wikipedia.org/wiki/Quartile">四分位</a> 和 <a class="reference external" href="https://en.wikipedia.org/wiki/Decile">十分位</a>：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">round</span><span class="p">,</span> <span class="n">sat</span><span class="o">.</span><span class="n">quantiles</span><span class="p">()))</span>
|
||
<span class="go">[928, 1060, 1192]</span>
|
||
<span class="gp">>>> </span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">round</span><span class="p">,</span> <span class="n">sat</span><span class="o">.</span><span class="n">quantiles</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">10</span><span class="p">)))</span>
|
||
<span class="go">[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>为了估算一个不易解析的模型分布，<a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 可以生成用于 <a class="reference external" href="https://en.wikipedia.org/wiki/Monte_Carlo_method">蒙特卡洛模拟</a> 的输入样本：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">model</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
|
||
<span class="gp">... </span> <span class="k">return</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">11</span> <span class="o">*</span> <span class="n">z</span><span class="p">)</span>
|
||
<span class="gp">...</span>
|
||
<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">100_000</span>
|
||
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">3652260728</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">Y</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">4582495471</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">Z</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">6582483453</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">quantiles</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">))</span>
|
||
<span class="go">[1.4591308524824727, 1.8035946855390597, 2.175091447274739]</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>当样本量较大并且成功试验的可能性接近 50% 时，正态分布可以被用来模拟 <a class="reference external" href="http://mathworld.wolfram.com/BinomialDistribution.html">二项分布</a>。</p>
|
||
<p>例如，一次开源会议有 750 名与会者和两个可分别容纳 500 人的会议厅。 会上有一场关于 Python 的演讲和一场关于 Ruby 的演讲。 在往届会议中，65% 的与会者更愿意去听关于 Python 的演讲。 假定人群的偏好没有发生改变，那么 Python 演讲的会议厅不超出其容量上限的可能性是多少？</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">750</span> <span class="c1"># Sample size</span>
|
||
<span class="gp">>>> </span><span class="n">p</span> <span class="o">=</span> <span class="mf">0.65</span> <span class="c1"># Preference for Python</span>
|
||
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="n">p</span> <span class="c1"># Preference for Ruby</span>
|
||
<span class="gp">>>> </span><span class="n">k</span> <span class="o">=</span> <span class="mi">500</span> <span class="c1"># Room capacity</span>
|
||
|
||
<span class="gp">>>> </span><span class="c1"># Approximation using the cumulative normal distribution</span>
|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">sqrt</span>
|
||
<span class="gp">>>> </span><span class="nb">round</span><span class="p">(</span><span class="n">NormalDist</span><span class="p">(</span><span class="n">mu</span><span class="o">=</span><span class="n">n</span><span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">))</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span>
|
||
<span class="go">0.8402</span>
|
||
|
||
<span class="gp">>>> </span><span class="c1"># Solution using the cumulative binomial distribution</span>
|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">comb</span><span class="p">,</span> <span class="n">fsum</span>
|
||
<span class="gp">>>> </span><span class="nb">round</span><span class="p">(</span><span class="n">fsum</span><span class="p">(</span><span class="n">comb</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="n">r</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">4</span><span class="p">)</span>
|
||
<span class="go">0.8402</span>
|
||
|
||
<span class="gp">>>> </span><span class="c1"># Approximation using a simulation</span>
|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">seed</span><span class="p">,</span> <span class="n">choices</span>
|
||
<span class="gp">>>> </span><span class="n">seed</span><span class="p">(</span><span class="mi">8675309</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">trial</span><span class="p">():</span>
|
||
<span class="gp">... </span> <span class="k">return</span> <span class="n">choices</span><span class="p">((</span><span class="s1">'Python'</span><span class="p">,</span> <span class="s1">'Ruby'</span><span class="p">),</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s1">'Python'</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">mean</span><span class="p">(</span><span class="n">trial</span><span class="p">()</span> <span class="o"><=</span> <span class="n">k</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10_000</span><span class="p">))</span>
|
||
<span class="go">0.8398</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>在机器学习问题中也经常会出现正态分布。</p>
|
||
<p>Wikipedia 上有一个 <a class="reference external" href="https://en.wikipedia.org/wiki/Naive_Bayes_classifier#Sex_classification">朴素贝叶斯分类器的好例子</a>。 挑战的问题是根据对多个正态分布的特征测量值包括身高、体重和足部尺码来预测一个人的性别。</p>
|
||
<p>我们得到了由八个人的测量值组成的训练数据集。 假定这些测量值是正态分布的，因此我们用 <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> 来总结数据：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">height_male</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mf">5.92</span><span class="p">,</span> <span class="mf">5.58</span><span class="p">,</span> <span class="mf">5.92</span><span class="p">])</span>
|
||
<span class="gp">>>> </span><span class="n">height_female</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="mf">5.5</span><span class="p">,</span> <span class="mf">5.42</span><span class="p">,</span> <span class="mf">5.75</span><span class="p">])</span>
|
||
<span class="gp">>>> </span><span class="n">weight_male</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">180</span><span class="p">,</span> <span class="mi">190</span><span class="p">,</span> <span class="mi">170</span><span class="p">,</span> <span class="mi">165</span><span class="p">])</span>
|
||
<span class="gp">>>> </span><span class="n">weight_female</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">100</span><span class="p">,</span> <span class="mi">150</span><span class="p">,</span> <span class="mi">130</span><span class="p">,</span> <span class="mi">150</span><span class="p">])</span>
|
||
<span class="gp">>>> </span><span class="n">foot_size_male</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">12</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
|
||
<span class="gp">>>> </span><span class="n">foot_size_female</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</span><span class="p">])</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>接下来，我们遇到一个特征测量值已知但性别未知的新人：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">ht</span> <span class="o">=</span> <span class="mf">6.0</span> <span class="c1"># height</span>
|
||
<span class="gp">>>> </span><span class="n">wt</span> <span class="o">=</span> <span class="mi">130</span> <span class="c1"># weight</span>
|
||
<span class="gp">>>> </span><span class="n">fs</span> <span class="o">=</span> <span class="mi">8</span> <span class="c1"># foot size</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>从是男是女各 50% 的 <a class="reference external" href="https://en.wikipedia.org/wiki/Prior_probability">先验概率</a> 出发，我们通过将该先验概率乘以给定性别的特征度量值的可能性累积值来计算后验概率：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">prior_male</span> <span class="o">=</span> <span class="mf">0.5</span>
|
||
<span class="gp">>>> </span><span class="n">prior_female</span> <span class="o">=</span> <span class="mf">0.5</span>
|
||
<span class="gp">>>> </span><span class="n">posterior_male</span> <span class="o">=</span> <span class="p">(</span><span class="n">prior_male</span> <span class="o">*</span> <span class="n">height_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">*</span>
|
||
<span class="gp">... </span> <span class="n">weight_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">wt</span><span class="p">)</span> <span class="o">*</span> <span class="n">foot_size_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">fs</span><span class="p">))</span>
|
||
|
||
<span class="gp">>>> </span><span class="n">posterior_female</span> <span class="o">=</span> <span class="p">(</span><span class="n">prior_female</span> <span class="o">*</span> <span class="n">height_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">*</span>
|
||
<span class="gp">... </span> <span class="n">weight_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">wt</span><span class="p">)</span> <span class="o">*</span> <span class="n">foot_size_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">fs</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>最终预测值应为最大后验概率值。 这种算法被称为 <a class="reference external" href="https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation">maximum a posteriori</a> 或 MAP：</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="s1">'male'</span> <span class="k">if</span> <span class="n">posterior_male</span> <span class="o">></span> <span class="n">posterior_female</span> <span class="k">else</span> <span class="s1">'female'</span>
|
||
<span class="go">'female'</span>
|
||
</pre></div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
</section>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
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||
<div class="sphinxsidebarwrapper">
|
||
<h3><a href="../contents.html">目录</a></h3>
|
||
<ul>
|
||
<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code> --- 数学统计函数</a><ul>
|
||
<li><a class="reference internal" href="#averages-and-measures-of-central-location">平均值以及对中心位置的评估</a></li>
|
||
<li><a class="reference internal" href="#measures-of-spread">对分散程度的评估</a></li>
|
||
<li><a class="reference internal" href="#function-details">函数细节</a></li>
|
||
<li><a class="reference internal" href="#exceptions">异常</a></li>
|
||
<li><a class="reference internal" href="#normaldist-objects"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code> 对象</a><ul>
|
||
<li><a class="reference internal" href="#normaldist-examples-and-recipes"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code> 示例和用法</a></li>
|
||
</ul>
|
||
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|
||
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|
||
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|
||
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