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  <p class="right" id="statisticsp-0"><span id="index-0"></span><strong>protocol</strong></p>
<section id="statisticsp">
<h1><code class="docutils literal notranslate"><span class="pre">statisticsp</span></code><a class="headerlink" href="#statisticsp" title="Link to this heading"></a></h1>
<p>Statistical calculations over a list of numbers protocol.</p>
<div class="line-block">
<div class="line"><strong>Availability:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">logtalk_load(statistics(loader))</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Author:</strong> Paulo Moura</div>
<div class="line"><strong>Version:</strong> 1:3:0</div>
<div class="line"><strong>Date:</strong> 2022-06-20</div>
</div>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Dependencies:</strong></div>
<div class="line-block">
<div class="line">(none)</div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Remarks:</strong></div>
<div class="line-block">
<div class="line">(none)</div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Inherited public predicates:</strong></div>
<div class="line-block">
<div class="line">(none)</div>
</div>
</div>
<nav class="contents local" id="contents">
<ul class="simple">
<li><p><a class="reference internal" href="#public-predicates" id="id1">Public predicates</a></p>
<ul>
<li><p><a class="reference internal" href="#product-2" id="id2"><code class="docutils literal notranslate"><span class="pre">product/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#sum-2" id="id3"><code class="docutils literal notranslate"><span class="pre">sum/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#min-2" id="id4"><code class="docutils literal notranslate"><span class="pre">min/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#max-2" id="id5"><code class="docutils literal notranslate"><span class="pre">max/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#min-max-3" id="id6"><code class="docutils literal notranslate"><span class="pre">min_max/3</span></code></a></p></li>
<li><p><a class="reference internal" href="#range-2" id="id7"><code class="docutils literal notranslate"><span class="pre">range/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#arithmetic-mean-2" id="id8"><code class="docutils literal notranslate"><span class="pre">arithmetic_mean/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#geometric-mean-2" id="id9"><code class="docutils literal notranslate"><span class="pre">geometric_mean/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#harmonic-mean-2" id="id10"><code class="docutils literal notranslate"><span class="pre">harmonic_mean/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#weighted-mean-3" id="id11"><code class="docutils literal notranslate"><span class="pre">weighted_mean/3</span></code></a></p></li>
<li><p><a class="reference internal" href="#median-2" id="id12"><code class="docutils literal notranslate"><span class="pre">median/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#modes-2" id="id13"><code class="docutils literal notranslate"><span class="pre">modes/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#average-deviation-3" id="id14"><code class="docutils literal notranslate"><span class="pre">average_deviation/3</span></code></a></p></li>
<li><p><a class="reference internal" href="#mean-deviation-2" id="id15"><code class="docutils literal notranslate"><span class="pre">mean_deviation/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#median-deviation-2" id="id16"><code class="docutils literal notranslate"><span class="pre">median_deviation/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#standard-deviation-2" id="id17"><code class="docutils literal notranslate"><span class="pre">standard_deviation/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#coefficient-of-variation-2" id="id18"><code class="docutils literal notranslate"><span class="pre">coefficient_of_variation/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#relative-standard-deviation-2" id="id19"><code class="docutils literal notranslate"><span class="pre">relative_standard_deviation/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#skewness-2" id="id20"><code class="docutils literal notranslate"><span class="pre">skewness/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#kurtosis-2" id="id21"><code class="docutils literal notranslate"><span class="pre">kurtosis/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#variance-2" id="id22"><code class="docutils literal notranslate"><span class="pre">variance/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#z-normalization-2" id="id23"><code class="docutils literal notranslate"><span class="pre">z_normalization/2</span></code></a></p></li>
<li><p><a class="reference internal" href="#fractile-3" id="id24"><code class="docutils literal notranslate"><span class="pre">fractile/3</span></code></a></p></li>
<li><p><a class="reference internal" href="#valid-1" id="id25"><code class="docutils literal notranslate"><span class="pre">valid/1</span></code></a></p></li>
</ul>
</li>
<li><p><a class="reference internal" href="#protected-predicates" id="id26">Protected predicates</a></p></li>
<li><p><a class="reference internal" href="#private-predicates" id="id27">Private predicates</a></p></li>
<li><p><a class="reference internal" href="#operators" id="id28">Operators</a></p></li>
</ul>
</nav>
<section id="public-predicates">
<h2><a class="toc-backref" href="#contents" role="doc-backlink">Public predicates</a><a class="headerlink" href="#public-predicates" title="Link to this heading"></a></h2>
<section id="product-2">
<span id="statisticsp-0-product-2"></span><span id="index-1"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">product/2</span></code></a><a class="headerlink" href="#product-2" title="Link to this heading"></a></h3>
<p>Calculates the product of all list numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">product(List,Product)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">product(+list(number),-number)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="sum-2">
<span id="statisticsp-0-sum-2"></span><span id="index-2"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">sum/2</span></code></a><a class="headerlink" href="#sum-2" title="Link to this heading"></a></h3>
<p>Calculates the sum of all list numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">sum(List,Sum)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">sum(+list(number),-number)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="min-2">
<span id="statisticsp-0-min-2"></span><span id="index-3"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">min/2</span></code></a><a class="headerlink" href="#min-2" title="Link to this heading"></a></h3>
<p>Determines the minimum value in a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">min(List,Minimum)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">min(+list,-number)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="max-2">
<span id="statisticsp-0-max-2"></span><span id="index-4"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">max/2</span></code></a><a class="headerlink" href="#max-2" title="Link to this heading"></a></h3>
<p>Determines the list maximum value in a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">max(List,Maximum)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">max(+list,-number)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="min-max-3">
<span id="statisticsp-0-min-max-3"></span><span id="index-5"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">min_max/3</span></code></a><a class="headerlink" href="#min-max-3" title="Link to this heading"></a></h3>
<p>Determines the minimum and maximum values in a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">min_max(List,Minimum,Maximum)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">min_max(+list(number),-number,-number)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="range-2">
<span id="statisticsp-0-range-2"></span><span id="index-6"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">range/2</span></code></a><a class="headerlink" href="#range-2" title="Link to this heading"></a></h3>
<p>Range is the length of the smallest interval which contains all the numbers in List. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">range(List,Range)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">range(+list,-number)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="arithmetic-mean-2">
<span id="statisticsp-0-arithmetic-mean-2"></span><span id="index-7"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">arithmetic_mean/2</span></code></a><a class="headerlink" href="#arithmetic-mean-2" title="Link to this heading"></a></h3>
<p>Calculates the arithmetic mean of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">arithmetic_mean(List,Mean)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">arithmetic_mean(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="geometric-mean-2">
<span id="statisticsp-0-geometric-mean-2"></span><span id="index-8"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">geometric_mean/2</span></code></a><a class="headerlink" href="#geometric-mean-2" title="Link to this heading"></a></h3>
<p>Calculates the geometric mean of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">geometric_mean(List,Mean)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">geometric_mean(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="harmonic-mean-2">
<span id="statisticsp-0-harmonic-mean-2"></span><span id="index-9"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">harmonic_mean/2</span></code></a><a class="headerlink" href="#harmonic-mean-2" title="Link to this heading"></a></h3>
<p>Calculates the harmonic mean of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">harmonic_mean(List,Mean)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">harmonic_mean(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="weighted-mean-3">
<span id="statisticsp-0-weighted-mean-3"></span><span id="index-10"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">weighted_mean/3</span></code></a><a class="headerlink" href="#weighted-mean-3" title="Link to this heading"></a></h3>
<p>Calculates the weighted mean of a list of numbers. Fails if the list is empty or if the two lists have different lengths. Wights are assume to be non-negative.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">weighted_mean(Weights,List,Mean)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">weighted_mean(+list(number),+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="median-2">
<span id="statisticsp-0-median-2"></span><span id="index-11"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">median/2</span></code></a><a class="headerlink" href="#median-2" title="Link to this heading"></a></h3>
<p>Calculates the median of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">median(List,Median)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">median(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="modes-2">
<span id="statisticsp-0-modes-2"></span><span id="index-12"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">modes/2</span></code></a><a class="headerlink" href="#modes-2" title="Link to this heading"></a></h3>
<p>Returns the list of modes of a list of numbers in ascending order. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">modes(List,Modes)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">modes(+list(number),-list(number))</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="average-deviation-3">
<span id="statisticsp-0-average-deviation-3"></span><span id="index-13"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">average_deviation/3</span></code></a><a class="headerlink" href="#average-deviation-3" title="Link to this heading"></a></h3>
<p>Calculates the average absolute deviation of a list of numbers given a central tendency (e.g., mean, median, or mode). Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">average_deviation(List,CentralTendency,Deviation)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">average_deviation(+list(number),+float,-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="mean-deviation-2">
<span id="statisticsp-0-mean-deviation-2"></span><span id="index-14"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">mean_deviation/2</span></code></a><a class="headerlink" href="#mean-deviation-2" title="Link to this heading"></a></h3>
<p>Calculates the mean absolute deviation of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">mean_deviation(List,Deviation)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">mean_deviation(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="median-deviation-2">
<span id="statisticsp-0-median-deviation-2"></span><span id="index-15"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">median_deviation/2</span></code></a><a class="headerlink" href="#median-deviation-2" title="Link to this heading"></a></h3>
<p>Calculates the median absolute deviation of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">median_deviation(List,Deviation)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">median_deviation(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="standard-deviation-2">
<span id="statisticsp-0-standard-deviation-2"></span><span id="index-16"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">standard_deviation/2</span></code></a><a class="headerlink" href="#standard-deviation-2" title="Link to this heading"></a></h3>
<p>Calculates the standard deviation of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">standard_deviation(List,Deviation)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">standard_deviation(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="coefficient-of-variation-2">
<span id="statisticsp-0-coefficient-of-variation-2"></span><span id="index-17"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">coefficient_of_variation/2</span></code></a><a class="headerlink" href="#coefficient-of-variation-2" title="Link to this heading"></a></h3>
<p>Calculates the coefficient of variation of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">coefficient_of_variation(List,Coefficient)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">coefficient_of_variation(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="relative-standard-deviation-2">
<span id="statisticsp-0-relative-standard-deviation-2"></span><span id="index-18"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">relative_standard_deviation/2</span></code></a><a class="headerlink" href="#relative-standard-deviation-2" title="Link to this heading"></a></h3>
<p>Calculates the relative standard deviation of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">relative_standard_deviation(List,Percentage)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">relative_standard_deviation(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="skewness-2">
<span id="statisticsp-0-skewness-2"></span><span id="index-19"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">skewness/2</span></code></a><a class="headerlink" href="#skewness-2" title="Link to this heading"></a></h3>
<p>Calculates the (moment) skewness of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">skewness(List,Skewness)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">skewness(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="kurtosis-2">
<span id="statisticsp-0-kurtosis-2"></span><span id="index-20"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">kurtosis/2</span></code></a><a class="headerlink" href="#kurtosis-2" title="Link to this heading"></a></h3>
<p>Calculates the (excess) kurtosis of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">kurtosis(List,Kurtosis)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">kurtosis(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="variance-2">
<span id="statisticsp-0-variance-2"></span><span id="index-21"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">variance/2</span></code></a><a class="headerlink" href="#variance-2" title="Link to this heading"></a></h3>
<p>Calculates the unbiased variance of a list of numbers. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">variance(List,Variance)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">variance(+list(number),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="z-normalization-2">
<span id="statisticsp-0-z-normalization-2"></span><span id="index-22"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">z_normalization/2</span></code></a><a class="headerlink" href="#z-normalization-2" title="Link to this heading"></a></h3>
<p>Normalizes a list of number such that for the resulting list the mean of is close to zero and the standard deviation is close to 1. Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">z_normalization(List,NormalizedList)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">z_normalization(+list(number),-list(float))</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="fractile-3">
<span id="statisticsp-0-fractile-3"></span><span id="index-23"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">fractile/3</span></code></a><a class="headerlink" href="#fractile-3" title="Link to this heading"></a></h3>
<p>Calculates the smallest value in a list of numbers such that the list elements in its fraction <code class="docutils literal notranslate"><span class="pre">P</span></code> are less or equal to that value (with <code class="docutils literal notranslate"><span class="pre">P</span></code> in the open interval <code class="docutils literal notranslate"><span class="pre">(0.0,</span> <span class="pre">1.0)</span></code>). Fails if the list is empty.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">fractile(P,List,Fractile)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">fractile(+float,+list(integer),-integer)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
<div class="line"><code class="docutils literal notranslate"><span class="pre">fractile(+float,+list(float),-float)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
<hr class="docutils" />
</section>
<section id="valid-1">
<span id="statisticsp-0-valid-1"></span><span id="index-24"></span><h3><a class="toc-backref" href="#contents" role="doc-backlink"><code class="docutils literal notranslate"><span class="pre">valid/1</span></code></a><a class="headerlink" href="#valid-1" title="Link to this heading"></a></h3>
<p>Term is a closed list of numbers.</p>
<div class="line-block">
<div class="line"><strong>Compilation flags:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">static</span></code></div>
</div>
</div>
<div class="line-block">
<div class="line"><strong>Template:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">valid(Term)</span></code></div>
</div>
<div class="line"><strong>Mode and number of proofs:</strong></div>
<div class="line-block">
<div class="line"><code class="docutils literal notranslate"><span class="pre">valid(&#64;nonvar)</span></code> - <code class="docutils literal notranslate"><span class="pre">zero_or_one</span></code></div>
</div>
</div>
</section>
</section>
<hr class="docutils" />
<section id="protected-predicates">
<h2><a class="toc-backref" href="#contents" role="doc-backlink">Protected predicates</a><a class="headerlink" href="#protected-predicates" title="Link to this heading"></a></h2>
<p>(none)</p>
</section>
<section id="private-predicates">
<h2><a class="toc-backref" href="#contents" role="doc-backlink">Private predicates</a><a class="headerlink" href="#private-predicates" title="Link to this heading"></a></h2>
<p>(none)</p>
</section>
<section id="operators">
<h2><a class="toc-backref" href="#contents" role="doc-backlink">Operators</a><a class="headerlink" href="#operators" title="Link to this heading"></a></h2>
<p>(none)</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="statistics_0.html#statistics-0"><span class="std std-ref">statistics</span></a>, <a class="reference internal" href="sample_0.html#sample-0"><span class="std std-ref">sample</span></a>, <a class="reference internal" href="population_0.html#population-0"><span class="std std-ref">population</span></a></p>
</div>
</section>
</section>


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