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<b class="current">The <tt class="ttfamily">stats</tt> module</b>
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<div><h2 id="a0000000254">2.9 The <tt class="ttfamily">stats</tt> module</h2>
<p>The <tt class="ttfamily">stats</tt> module contains statistical functions: </p><p>   <big class="large"><b class="bf">stats.binomialCDF(<img src="images/img-0719.png" alt="$k,p,n$" style="vertical-align:-4px; 
                                     width:45px; 
                                     height:17px" class="math gen" />)</b></big> <br />The stats.binomialCDF(<img src="images/img-0719.png" alt="$k,p,n$" style="vertical-align:-4px; 
                                     width:45px; 
                                     height:17px" class="math gen" />) function evaluates the probability of getting fewer than or exactly <img src="images/img-0654.png" alt="$k$" style="vertical-align:0px; 
                                     width:8px; 
                                     height:13px" class="math gen" /> successes out of <img src="images/img-0025.png" alt="$n$" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> trials in a binomial distribution with success probability <img src="images/img-0270.png" alt="$p$" style="vertical-align:-4px; 
                                     width:10px; 
                                     height:12px" class="math gen" />. <img src="images/img-0654.png" alt="$k$" style="vertical-align:0px; 
                                     width:8px; 
                                     height:13px" class="math gen" /> and <img src="images/img-0025.png" alt="$n$" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must be positive real integers. <img src="images/img-0270.png" alt="$p$" style="vertical-align:-4px; 
                                     width:10px; 
                                     height:12px" class="math gen" /> must be a real number in the range <img src="images/img-0720.png" alt="$0\leq p \leq 1$" style="vertical-align:-4px; 
                                     width:74px; 
                                     height:16px" class="math gen" />. <a name="a0000001601" id="a0000001601"></a>  </p><p>    <big class="large"><b class="bf">stats.binomialPDF(<img src="images/img-0719.png" alt="$k,p,n$" style="vertical-align:-4px; 
                                     width:45px; 
                                     height:17px" class="math gen" />)</b></big> <br />The stats.binomialPDF(<img src="images/img-0719.png" alt="$k,p,n$" style="vertical-align:-4px; 
                                     width:45px; 
                                     height:17px" class="math gen" />) function evaluates the probability of getting <img src="images/img-0654.png" alt="$k$" style="vertical-align:0px; 
                                     width:8px; 
                                     height:13px" class="math gen" /> successes out of <img src="images/img-0025.png" alt="$n$" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> trials in a binomial distribution with success probability <img src="images/img-0270.png" alt="$p$" style="vertical-align:-4px; 
                                     width:10px; 
                                     height:12px" class="math gen" />. <img src="images/img-0654.png" alt="$k$" style="vertical-align:0px; 
                                     width:8px; 
                                     height:13px" class="math gen" /> and <img src="images/img-0025.png" alt="$n$" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must be positive real integers. <img src="images/img-0270.png" alt="$p$" style="vertical-align:-4px; 
                                     width:10px; 
                                     height:12px" class="math gen" /> must be a real number in the range <img src="images/img-0720.png" alt="$0\leq p \leq 1$" style="vertical-align:-4px; 
                                     width:74px; 
                                     height:16px" class="math gen" />. <a name="a0000001602" id="a0000001602"></a>  </p><p>    <big class="large"><b class="bf">stats.chisqCDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.chisqCDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />) function returns the cumulative probability density at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> in a <img src="images/img-0272.png" alt="$\chi $" style="vertical-align:-4px; 
                                     width:11px; 
                                     height:12px" class="math gen" />-squared distribution with <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> degrees of freedom. <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless integer. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless number. <a name="a0000001603" id="a0000001603"></a>  </p><p>    <big class="large"><b class="bf">stats.chisqCDFi(<img src="images/img-0722.png" alt="$P,\nu $" style="vertical-align:-4px; 
                                     width:30px; 
                                     height:16px" class="math gen" />)</b></big> <br />The stats.chisqCDFi(<img src="images/img-0722.png" alt="$P,\nu $" style="vertical-align:-4px; 
                                     width:30px; 
                                     height:16px" class="math gen" />) function returns the point <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> at which the cumulative probability density in a <img src="images/img-0272.png" alt="$\chi $" style="vertical-align:-4px; 
                                     width:11px; 
                                     height:12px" class="math gen" />-squared distribution with <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> degrees of freedom is <img src="images/img-0723.png" alt="$P$" style="vertical-align:0px; 
                                     width:14px; 
                                     height:12px" class="math gen" />. <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless integer. <img src="images/img-0723.png" alt="$P$" style="vertical-align:0px; 
                                     width:14px; 
                                     height:12px" class="math gen" /> must be a real number in the range <img src="images/img-0720.png" alt="$0\leq p \leq 1$" style="vertical-align:-4px; 
                                     width:74px; 
                                     height:16px" class="math gen" />. <a name="a0000001604" id="a0000001604"></a>  </p><p>    <big class="large"><b class="bf">stats.chisqPDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.chisqPDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />) function returns the probability density at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> in a <img src="images/img-0272.png" alt="$\chi $" style="vertical-align:-4px; 
                                     width:11px; 
                                     height:12px" class="math gen" />-squared distribution with <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> degrees of freedom. <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless integer. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless number. <a name="a0000001605" id="a0000001605"></a>  </p><p>    <big class="large"><b class="bf">stats.gaussianCDF(<img src="images/img-0701.png" alt="$x,\sigma $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.gaussianCDF(<img src="images/img-0701.png" alt="$x,\sigma $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />) function evaluates the Gaussian cumulative distribution function of standard deviation <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" />. The distribution is centred upon <img src="images/img-0135.png" alt="$x=0$" style="vertical-align:0px; 
                                     width:43px; 
                                     height:12px" class="math gen" />. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> and <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must both be real, but may have any physical dimensions so long as they match. <a name="a0000001606" id="a0000001606"></a>  </p><p>    <big class="large"><b class="bf">stats.gaussianCDFi(<img src="images/img-0701.png" alt="$x,\sigma $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.gaussianCDFi(<img src="images/img-0701.png" alt="$x,\sigma $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />) function evaluates the inverse Gaussian cumulative distribution function of standard deviation <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" />. The distribution is centred upon <img src="images/img-0135.png" alt="$x=0$" style="vertical-align:0px; 
                                     width:43px; 
                                     height:12px" class="math gen" />. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> and <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must both be real, but may have any physical dimensions so long as they match. <a name="a0000001607" id="a0000001607"></a>  </p><p>    <big class="large"><b class="bf">stats.gaussianPDF(<img src="images/img-0701.png" alt="$x,\sigma $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.gaussianPDF(<img src="images/img-0701.png" alt="$x,\sigma $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />) function evaluates the Gaussian probability density function of standard deviation <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" />. The distribution is centred upon <img src="images/img-0135.png" alt="$x=0$" style="vertical-align:0px; 
                                     width:43px; 
                                     height:12px" class="math gen" />. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> and <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must both be real, but may have any physical dimensions so long as they match. <a name="a0000001608" id="a0000001608"></a>  </p><p>    <big class="large"><b class="bf">stats.lognormalCDF(<img src="images/img-0724.png" alt="$x,\zeta ,\sigma $" style="vertical-align:-4px; 
                                     width:46px; 
                                     height:17px" class="math gen" />)</b></big> <br />The stats.lognormalCDF(<img src="images/img-0724.png" alt="$x,\zeta ,\sigma $" style="vertical-align:-4px; 
                                     width:46px; 
                                     height:17px" class="math gen" />) function evaluates the log normal cumulative distribution function of standard deviation <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" />, centred upon <img src="images/img-0275.png" alt="$\zeta $" style="vertical-align:-4px; 
                                     width:9px; 
                                     height:17px" class="math gen" />, at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" />. <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must be real, positive and dimensionless. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> and <img src="images/img-0275.png" alt="$\zeta $" style="vertical-align:-4px; 
                                     width:9px; 
                                     height:17px" class="math gen" /> must both be real, but may have any physical dimensions so long as they match. <a name="a0000001609" id="a0000001609"></a>  </p><p>    <big class="large"><b class="bf">stats.lognormalCDFi(<img src="images/img-0724.png" alt="$x,\zeta ,\sigma $" style="vertical-align:-4px; 
                                     width:46px; 
                                     height:17px" class="math gen" />)</b></big> <br />The stats.lognormalCDFi(<img src="images/img-0724.png" alt="$x,\zeta ,\sigma $" style="vertical-align:-4px; 
                                     width:46px; 
                                     height:17px" class="math gen" />) function evaluates the inverse log normal cumulative distribution function of standard deviation <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" />, centred upon <img src="images/img-0275.png" alt="$\zeta $" style="vertical-align:-4px; 
                                     width:9px; 
                                     height:17px" class="math gen" />, at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" />. <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must be real, positive and dimensionless. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> and <img src="images/img-0275.png" alt="$\zeta $" style="vertical-align:-4px; 
                                     width:9px; 
                                     height:17px" class="math gen" /> must both be real, but may have any physical dimensions so long as they match. <a name="a0000001610" id="a0000001610"></a>  </p><p>    <big class="large"><b class="bf">stats.lognormalPDF(<img src="images/img-0724.png" alt="$x,\zeta ,\sigma $" style="vertical-align:-4px; 
                                     width:46px; 
                                     height:17px" class="math gen" />)</b></big> <br />The stats.lognormalPDF(<img src="images/img-0724.png" alt="$x,\zeta ,\sigma $" style="vertical-align:-4px; 
                                     width:46px; 
                                     height:17px" class="math gen" />) function evaluates the log normal probability density function of standard deviation <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" />, centred upon <img src="images/img-0275.png" alt="$\zeta $" style="vertical-align:-4px; 
                                     width:9px; 
                                     height:17px" class="math gen" />, at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" />. <img src="images/img-0273.png" alt="$\sigma $" style="vertical-align:0px; 
                                     width:11px; 
                                     height:8px" class="math gen" /> must be real, positive and dimensionless. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> and <img src="images/img-0275.png" alt="$\zeta $" style="vertical-align:-4px; 
                                     width:9px; 
                                     height:17px" class="math gen" /> must both be real, but may have any physical dimensions so long as they match. <a name="a0000001611" id="a0000001611"></a>  </p><p>    <big class="large"><b class="bf">stats.poissonCDF(<img src="images/img-0725.png" alt="$x,\mu $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.poissonCDF(<img src="images/img-0725.png" alt="$x,\mu $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />) function returns the probability of getting <img src="images/img-0726.png" alt="$\leq x$" style="vertical-align:-3px; 
                                     width:28px; 
                                     height:15px" class="math gen" /> from a Poisson distribution with mean <img src="images/img-0727.png" alt="$\mu $" style="vertical-align:-4px; 
                                     width:11px; 
                                     height:12px" class="math gen" />, where <img src="images/img-0727.png" alt="$\mu $" style="vertical-align:-4px; 
                                     width:11px; 
                                     height:12px" class="math gen" /> must be real, positive and dimensionless and <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> must be real and dimensionless. <a name="a0000001612" id="a0000001612"></a>  </p><p>    <big class="large"><b class="bf">stats.poissonPDF(<img src="images/img-0725.png" alt="$x,\mu $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.poissonPDF(<img src="images/img-0725.png" alt="$x,\mu $" style="vertical-align:-4px; 
                                     width:29px; 
                                     height:12px" class="math gen" />) function returns the probability of getting <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> from a Poisson distribution with mean <img src="images/img-0727.png" alt="$\mu $" style="vertical-align:-4px; 
                                     width:11px; 
                                     height:12px" class="math gen" />, where <img src="images/img-0727.png" alt="$\mu $" style="vertical-align:-4px; 
                                     width:11px; 
                                     height:12px" class="math gen" /> must be real, positive and dimensionless and <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> must be a real dimensionless integer. <a name="a0000001613" id="a0000001613"></a>  </p><p>    <big class="large"><b class="bf">stats.tdistCDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.tdistCDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />) function returns the cumulative probability density at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> in a <img src="images/img-0065.png" alt="$t$" style="vertical-align:0px; 
                                     width:6px; 
                                     height:12px" class="math gen" />-distribution with <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> degrees of freedom. <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless integer. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless number. <a name="a0000001614" id="a0000001614"></a>  </p><p>    <big class="large"><b class="bf">stats.tdistCDFi(<img src="images/img-0722.png" alt="$P,\nu $" style="vertical-align:-4px; 
                                     width:30px; 
                                     height:16px" class="math gen" />)</b></big> <br />The stats.tdistCDFi(<img src="images/img-0722.png" alt="$P,\nu $" style="vertical-align:-4px; 
                                     width:30px; 
                                     height:16px" class="math gen" />) function returns the point <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> at which the cumulative probability density in a <img src="images/img-0065.png" alt="$t$" style="vertical-align:0px; 
                                     width:6px; 
                                     height:12px" class="math gen" />-distribution with <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> degrees of freedom is <img src="images/img-0723.png" alt="$P$" style="vertical-align:0px; 
                                     width:14px; 
                                     height:12px" class="math gen" />. <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless integer. <img src="images/img-0723.png" alt="$P$" style="vertical-align:0px; 
                                     width:14px; 
                                     height:12px" class="math gen" /> must be a real number in the range <img src="images/img-0720.png" alt="$0\leq p \leq 1$" style="vertical-align:-4px; 
                                     width:74px; 
                                     height:16px" class="math gen" />. <a name="a0000001615" id="a0000001615"></a>  </p><p>    <big class="large"><b class="bf">stats.tdistPDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />)</b></big> <br />The stats.tdistPDF(<img src="images/img-0721.png" alt="$x,\nu $" style="vertical-align:-4px; 
                                     width:28px; 
                                     height:12px" class="math gen" />) function returns the probability density at <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> in a <img src="images/img-0065.png" alt="$t$" style="vertical-align:0px; 
                                     width:6px; 
                                     height:12px" class="math gen" />-distribution with <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> degrees of freedom. <img src="images/img-0271.png" alt="$\nu $" style="vertical-align:0px; 
                                     width:9px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless integer. <img src="images/img-0030.png" alt="$x$" style="vertical-align:0px; 
                                     width:10px; 
                                     height:8px" class="math gen" /> must be a positive real dimensionless number. <a name="a0000001616" id="a0000001616"></a>  </p></div>





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