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<h1>Relay Races and Revolving Doors</h1>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#are-you-popular-hint-no">Are you popular? Hint: no.</a></li>
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<h1>Relay Races and Revolving Doors<a class="headerlink" href="#relay-races-and-revolving-doors" title="Permalink to this heading">#</a></h1>
<p><a class="reference external" href="https://colab.research.google.com/github/AllenDowney/ProbablyOverthinkingIt/blob/book/notebooks/inspection.ipynb">Click here to run this notebook on Colab</a>.</p>
<p>I gave a talk related to this chapter at PyData NYC 2019. You can <a class="reference external" href="https://www.youtube.com/watch?v=cXWTHfvycyM">watch the video here</a>.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Install empirical dist if we don't already have it</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">empiricaldist</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="o">!</span>pip<span class="w"> </span>install<span class="w"> </span>empiricaldist
</pre></div>
</div>
</div>
</details>
</div>
<div class="cell tag_hide-cell docutils container">
<details class="hide above-input">
<summary aria-label="Toggle hidden content">
<span class="collapsed">Show code cell content</span>
<span class="expanded">Hide code cell content</span>
</summary>
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># download utils.py</span>
<span class="kn">from</span> <span class="nn">os.path</span> <span class="kn">import</span> <span class="n">basename</span><span class="p">,</span> <span class="n">exists</span>
<span class="k">def</span> <span class="nf">download</span><span class="p">(</span><span class="n">url</span><span class="p">):</span>
<span class="n">filename</span> <span class="o">=</span> <span class="n">basename</span><span class="p">(</span><span class="n">url</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">exists</span><span class="p">(</span><span class="n">filename</span><span class="p">):</span>
<span class="kn">from</span> <span class="nn">urllib.request</span> <span class="kn">import</span> <span class="n">urlretrieve</span>
<span class="n">local</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">urlretrieve</span><span class="p">(</span><span class="n">url</span><span class="p">,</span> <span class="n">filename</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'Downloaded '</span> <span class="o">+</span> <span class="n">local</span><span class="p">)</span>
<span class="n">download</span><span class="p">(</span><span class="s2">"https://github.com/AllenDowney/ProbablyOverthinkingIt/raw/book/notebooks/utils.py"</span><span class="p">)</span>
</pre></div>
</div>
</div>
</details>
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<details class="hide above-input">
<summary aria-label="Toggle hidden content">
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</summary>
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="c1"># Set the random seed so we get the same results every time</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">17</span><span class="p">)</span>
</pre></div>
</div>
</div>
</details>
</div>
<p>When you run 209 miles, you have a lot of time to think. In 2010, I was
a member of a 12-person team that ran a 209-mile relay race in New Hampshire.</p>
<p>Long-distance relay races are an unusual format, and this was the first
(and last!) time I participated in one. I ran the third leg, so when I
joined the race, it had been going for a few hours and runners were
spread out over several miles of the course.</p>
<p>After I ran a few miles, I noticed something unusual:</p>
<ul class="simple">
<li><p>There were more fast runners in the race than I expected. Several times I was overtaken by runners much faster than me.</p></li>
<li><p>There were also more slow runners than I expected. When I passed other runners, I was often much faster than them.</p></li>
</ul>
<p>At first I thought this pattern might reflect the kind of people who
sign up for a 209-mile race. Maybe, for some reason, this format appeals
primarily to runners who are much faster than average or much slower,
and not as much to middle-of-the-pack runners like me.</p>
<p>After the race, with more oxygen available to my brain, I realized that
this explanation is wrong. To my embarrassment, I was fooled by a common
statistical error, one that I teach students in my classes!</p>
<p>The error is called length-biased sampling, and its effect is called the
inspection paradox. If you have not heard of it, this chapter will
change your life, because once you learn about the inspection paradox,
you see it everywhere.</p>
<p>To explain it, I’ll start with simple examples and we will work our way
up. Some of the examples are fun, but some are more serious. For
example, length-biased sampling shows up in the criminal justice system
and distorts our perception of prison sentences and the risk of repeat
offenders.</p>
<p>But it’s not all bad news; if you are aware of the inspection paradox,
sometimes you can use it to measure indirectly quantities that would be
hard or impossible to measure directly. As an example, I’ll explain a
clever system used during the COVID pandemic to track infections and
identify superspreaders.</p>
<p>But let’s start with class sizes.</p>
<section id="class-size">
<h2>Class Size<a class="headerlink" href="#class-size" title="Permalink to this heading">#</a></h2>
<p>The class sizes in this section come from data reported by
Purdue University for undergraduate class sizes in the 2013–14 academic
year.</p>
<p><a class="reference external" href="https://www.purdue.edu/datadigest/2013-14/InstrStuLIfe/DistUGClasses.html">The data was originally posted here</a></p>
<p><a class="reference external" href="https://web.archive.org/web/20160415011613/https://www.purdue.edu/datadigest/2013-14/InstrStuLIfe/DistUGClasses.html">Now archived here</a></p>
<p>I typed in the number of classes in the row labeled “Total all classes”.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sizes</span> <span class="o">=</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="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">9</span><span class="p">),</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">19</span><span class="p">),</span> <span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">29</span><span class="p">),</span> <span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">39</span><span class="p">),</span> <span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">49</span><span class="p">),</span> <span class="p">(</span><span class="mi">50</span><span class="p">,</span> <span class="mi">99</span><span class="p">),</span> <span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">300</span><span class="p">)]</span>
<span class="n">counts</span> <span class="o">=</span> <span class="p">[</span><span class="mi">138</span><span class="p">,</span> <span class="mi">635</span><span class="p">,</span> <span class="mi">1788</span><span class="p">,</span> <span class="mi">1979</span><span class="p">,</span> <span class="mi">796</span><span class="p">,</span> <span class="mi">354</span><span class="p">,</span> <span class="mi">487</span><span class="p">,</span> <span class="mi">333</span><span class="p">]</span>
</pre></div>
</div>
</div>
</div>
<p>Since the class sizes are specified in ranges, I used the following function to generate uniform random samples within each range.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">generate_sample</span><span class="p">(</span><span class="n">sizes</span><span class="p">,</span> <span class="n">counts</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Generate a sample from a distribution.</span>
<span class="sd"> sizes: sequence of (low, high) pairs</span>
<span class="sd"> counts: sequence of integers</span>
<span class="sd"> returns: NumPy array</span>
<span class="sd"> """</span>
<span class="n">t</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="p">(</span><span class="n">low</span><span class="p">,</span> <span class="n">high</span><span class="p">),</span> <span class="n">count</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">sizes</span><span class="p">,</span> <span class="n">counts</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="n">low</span><span class="p">,</span> <span class="n">high</span><span class="p">,</span> <span class="n">count</span><span class="p">)</span>
<span class="n">sample</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="p">,</span> <span class="n">high</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">count</span><span class="p">)</span>
<span class="n">t</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">sample</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">unbiased</span> <span class="o">=</span> <span class="n">generate_sample</span><span class="p">(</span><span class="n">sizes</span><span class="p">,</span> <span class="n">counts</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1 1 138
2 9 635
10 19 1788
20 29 1979
30 39 796
40 49 354
50 99 487
100 300 333
</pre></div>
</div>
</div>
</div>
<p>The following function takes a sample and generates a new random sample, using the given weights.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">resample_weighted</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="n">weights</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Generate a biased sample.</span>
<span class="sd"> sample: NumPy array</span>
<span class="sd"> weights: NumPy array</span>
<span class="sd"> returns: NumPy array</span>
<span class="sd"> """</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sample</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">weights</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">weights</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">biased</span> <span class="o">=</span> <span class="n">resample_weighted</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Here’s the distribution of class sizes as given in the report.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">decorate</span><span class="p">,</span> <span class="n">kdeplot</span>
<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Reported by the Dean"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C1"</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">"Class size"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of class sizes"</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/650e12c4546f7975340b569492f891abc6347c4f1c8be66b2070166a2cdf21cd.png" src="_images/650e12c4546f7975340b569492f891abc6347c4f1c8be66b2070166a2cdf21cd.png" />
</div>
</div>
<p>The upper bound in this figure, 300, is just my guess. The original data
indicates how many classes are bigger than 100, but it doesn’t say how
much bigger. For this example, though, we don’t have to be too precise.</p>
<p>Now here’s what the distribution would look like if we sampled students.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Reported by the Dean"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C1"</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">biased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Reported by students"</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">"--"</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">"Class size"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of class sizes"</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/3112e48c9303b5f811552c5cc853ce721a81b71fd04dfb40b767b04c9e19abaa.png" src="_images/3112e48c9303b5f811552c5cc853ce721a81b71fd04dfb40b767b04c9e19abaa.png" />
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>34.611827956989245
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">biased</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>92.59815668202765
</pre></div>
</div>
</div>
</div>
</section>
<section id="unbiasing-the-data">
<h2>Unbiasing the Data<a class="headerlink" href="#unbiasing-the-data" title="Permalink to this heading">#</a></h2>
<p>Now suppose we collect a biased sample of 500 class sizes and apply the inverse weights to estimate the unbiased distribution.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sample</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">biased</span><span class="p">,</span> <span class="mi">500</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">reweighted</span> <span class="o">=</span> <span class="n">resample_weighted</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">sample</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Reported by the Dean"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C1"</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">biased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Reported by students"</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">"--"</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">reweighted</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Reweighted student sample"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C2"</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">":"</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">"Class size"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of class sizes"</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/1406f22111b6571e72a0589b3230bbd864049471b0ad87d6ef8d6bafc721fc6d.png" src="_images/1406f22111b6571e72a0589b3230bbd864049471b0ad87d6ef8d6bafc721fc6d.png" />
</div>
</div>
<p>If the estimate were perfect, the solid and dotted lines would be
identical. But with a limited sample size, we underestimate the number
of small classes by a little and overestimate the number of classes with
50-80 students. Nevertheless, it works pretty well.</p>
<p>This strategy works in other cases where the actual distribution is not
available, deliberately or not. If we can collect a good quality sample
from the biased distribution, we can approximate the actual distribution
by drawing a sample from the biased data. This process is an example of
weighted resampling. It’s “weighted” in the sense
that some items are given more weight than others, that is, more
probability of being sampled. And it’s called “resampling” because we’re
drawing a random sample from something that is itself a random sample.</p>
</section>
<section id="wheres-my-train">
<h2>Where’s My Train?<a class="headerlink" href="#wheres-my-train" title="Permalink to this heading">#</a></h2>
<p>I collected the following data from the Red Line, which is a
subway line in Boston, Massachusetts. The MBTA, which operates the Red
Line, provides a real-time data service, which I used to record the
arrival times for 70 trains between 4pm and 5pm over several days.
I don’t remember when, but I think it was in 2010.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">unbiased</span> <span class="o">=</span> <span class="p">[</span>
<span class="mf">428.0</span><span class="p">,</span>
<span class="mf">705.0</span><span class="p">,</span>
<span class="mf">407.0</span><span class="p">,</span>
<span class="mf">465.0</span><span class="p">,</span>
<span class="mf">433.0</span><span class="p">,</span>
<span class="mf">425.0</span><span class="p">,</span>
<span class="mf">204.0</span><span class="p">,</span>
<span class="mf">506.0</span><span class="p">,</span>
<span class="mf">143.0</span><span class="p">,</span>
<span class="mf">351.0</span><span class="p">,</span>
<span class="mf">450.0</span><span class="p">,</span>
<span class="mf">598.0</span><span class="p">,</span>
<span class="mf">464.0</span><span class="p">,</span>
<span class="mf">749.0</span><span class="p">,</span>
<span class="mf">341.0</span><span class="p">,</span>
<span class="mf">586.0</span><span class="p">,</span>
<span class="mf">754.0</span><span class="p">,</span>
<span class="mf">256.0</span><span class="p">,</span>
<span class="mf">378.0</span><span class="p">,</span>
<span class="mf">435.0</span><span class="p">,</span>
<span class="mf">176.0</span><span class="p">,</span>
<span class="mf">405.0</span><span class="p">,</span>
<span class="mf">360.0</span><span class="p">,</span>
<span class="mf">519.0</span><span class="p">,</span>
<span class="mf">648.0</span><span class="p">,</span>
<span class="mf">374.0</span><span class="p">,</span>
<span class="mf">483.0</span><span class="p">,</span>
<span class="mf">537.0</span><span class="p">,</span>
<span class="mf">578.0</span><span class="p">,</span>
<span class="mf">534.0</span><span class="p">,</span>
<span class="mf">577.0</span><span class="p">,</span>
<span class="mf">619.0</span><span class="p">,</span>
<span class="mf">538.0</span><span class="p">,</span>
<span class="mf">331.0</span><span class="p">,</span>
<span class="mf">186.0</span><span class="p">,</span>
<span class="mf">629.0</span><span class="p">,</span>
<span class="mf">193.0</span><span class="p">,</span>
<span class="mf">360.0</span><span class="p">,</span>
<span class="mf">660.0</span><span class="p">,</span>
<span class="mf">484.0</span><span class="p">,</span>
<span class="mf">512.0</span><span class="p">,</span>
<span class="mf">315.0</span><span class="p">,</span>
<span class="mf">457.0</span><span class="p">,</span>
<span class="mf">404.0</span><span class="p">,</span>
<span class="mf">740.0</span><span class="p">,</span>
<span class="mf">388.0</span><span class="p">,</span>
<span class="mf">357.0</span><span class="p">,</span>
<span class="mf">485.0</span><span class="p">,</span>
<span class="mf">567.0</span><span class="p">,</span>
<span class="mf">160.0</span><span class="p">,</span>
<span class="mf">428.0</span><span class="p">,</span>
<span class="mf">387.0</span><span class="p">,</span>
<span class="mf">901.0</span><span class="p">,</span>
<span class="mf">187.0</span><span class="p">,</span>
<span class="mf">622.0</span><span class="p">,</span>
<span class="mf">616.0</span><span class="p">,</span>
<span class="mf">585.0</span><span class="p">,</span>
<span class="mf">474.0</span><span class="p">,</span>
<span class="mf">442.0</span><span class="p">,</span>
<span class="mf">499.0</span><span class="p">,</span>
<span class="mf">437.0</span><span class="p">,</span>
<span class="mf">620.0</span><span class="p">,</span>
<span class="mf">351.0</span><span class="p">,</span>
<span class="mf">286.0</span><span class="p">,</span>
<span class="mf">373.0</span><span class="p">,</span>
<span class="mf">232.0</span><span class="p">,</span>
<span class="mf">393.0</span><span class="p">,</span>
<span class="mf">745.0</span><span class="p">,</span>
<span class="mf">636.0</span><span class="p">,</span>
<span class="mf">758.0</span><span class="p">,</span>
<span class="p">]</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">unbiased</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">unbiased</span><span class="p">)</span> <span class="o">/</span> <span class="mi">60</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">biased</span> <span class="o">=</span> <span class="n">resample_weighted</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Here’s the unbiased distribution.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">16.5</span><span class="p">,</span> <span class="mi">101</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Seen by MBTA"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C1"</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span>
<span class="n">xlabel</span><span class="o">=</span><span class="s2">"Time between trains (min)"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of time between trains"</span>
<span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/599e459e47a000a2b6ae2612bc4acb9b37ba710daeff2e95fbcea35e61934570.png" src="_images/599e459e47a000a2b6ae2612bc4acb9b37ba710daeff2e95fbcea35e61934570.png" />
</div>
</div>
<p>And here’s the biased distribution as seen by passengers.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">16.5</span><span class="p">,</span> <span class="mi">101</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Seen by MBTA"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C1"</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">biased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Seen by passengers"</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">"--"</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span>
<span class="n">xlabel</span><span class="o">=</span><span class="s2">"Time between trains (min)"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of time between trains"</span>
<span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/a56cf7a29ddc6362dbdd2cdd30ccb09a54c10821d36742c440be2030ab138d03.png" src="_images/a56cf7a29ddc6362dbdd2cdd30ccb09a54c10821d36742c440be2030ab138d03.png" />
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">biased</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(8.570714285714285, 7.7680952380952375)
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">biased</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">unbiased</span><span class="p">))</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">unbiased</span><span class="p">)</span> <span class="o">*</span> <span class="mi">100</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>10.332250352479615
</pre></div>
</div>
</div>
</div>
<p>In the actual distribution, the average time between trains is 7.8
minutes; in the biased distribution, as seen by a random passenger, it
is 9.2 minutes, about 20% longer.</p>
</section>
<section id="are-you-popular-hint-no">
<h2>Are you popular? Hint: no.<a class="headerlink" href="#are-you-popular-hint-no" title="Permalink to this heading">#</a></h2>
<p>To demonstrate the friendship paradox, I used data from a sample of about 4000
Facebook users from the <a class="reference external" href="http://snap.stanford.edu/">Stanford Network Analysis Project (SNAP)</a>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">download</span><span class="p">(</span><span class="s2">"https://snap.stanford.edu/data/facebook_combined.txt.gz"</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>The following function reads the Facebook data and builds a NetworkX graph.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="k">def</span> <span class="nf">read_graph</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Read a graph from a file.</span>
<span class="sd"> filename: string</span>
<span class="sd"> return: Graph</span>
<span class="sd"> """</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">Graph</span><span class="p">()</span>
<span class="n">array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">loadtxt</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span><span class="n">array</span><span class="p">)</span>
<span class="k">return</span> <span class="n">G</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fb</span> <span class="o">=</span> <span class="n">read_graph</span><span class="p">(</span><span class="s2">"facebook_combined.txt.gz"</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">fb</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">fb</span><span class="o">.</span><span class="n">edges</span><span class="p">())</span>
<span class="n">n</span><span class="p">,</span> <span class="n">m</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(4039, 88234)
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">unbiased</span> <span class="o">=</span> <span class="p">[</span><span class="n">fb</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">node</span><span class="p">)</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">fb</span><span class="p">]</span>
<span class="nb">len</span><span class="p">(</span><span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>4039
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1045
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">biased</span> <span class="o">=</span> <span class="n">resample_weighted</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">high</span> <span class="o">=</span> <span class="mi">250</span>
</pre></div>
</div>
</div>
</div>
<p>Here’s the unbiased distribution.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">high</span><span class="p">,</span> <span class="mi">101</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Random sample of people"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C1"</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span>
<span class="n">xlabel</span><span class="o">=</span><span class="s2">"Number of friends"</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of number of friends"</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/e94b675d6b737b03c83452e3066857412633b02c7cce4beb3327a9f894bdf76b.png" src="_images/e94b675d6b737b03c83452e3066857412633b02c7cce4beb3327a9f894bdf76b.png" />
</div>
</div>
<p>And here’s the biased distribution.
Compared to the unbiased distribution, people with more than 50 friends are
overrepresented; people with fewer friends are underrepresented.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">high</span><span class="p">,</span> <span class="mi">101</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Random sample of people"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"C1"</span><span class="p">)</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">biased</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="s2">"Random sample of friends"</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">"--"</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span>
<span class="n">xlabel</span><span class="o">=</span><span class="s2">"Number of friends"</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of number of friends"</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/4fff45e97f69885ebb22542574a230bff3cf89d70764a781a10660659950425b.png" src="_images/4fff45e97f69885ebb22542574a230bff3cf89d70764a781a10660659950425b.png" />
</div>
</div>
<p>The difference between the distributions is substantial: in the unbiased
sample, the average user has 44 friends; in the biased sample, the
average friend has 104, more than twice as many. And if you are a random
person in this sample, the probability that your friend is more popular
than you is about 76%.</p>
<p>If you are a bottlenose dolphin in Doubtful Sound, New Zealand, you interact regularly with between 1 and 12 other dolphins, according to the researchers who mapped your social network.
If I choose one of your “friends” at random and compare them to you, the probability is about 65% that your friend has more friends than you.</p>
<p>The dolphin data is from <a class="reference external" href="https://networkrepository.com/soc_dolphins.php">The Network Data Repository</a>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
<span class="n">download</span><span class="p">(</span><span class="s2">"http://nrvis.com/download/data/soc/soc-dolphins.zip"</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">exists</span><span class="p">(</span><span class="s1">'soc-dolphins.mtx'</span><span class="p">):</span>
<span class="o">!</span>unzip<span class="w"> </span>soc-dolphins.zip
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.io</span> <span class="kn">import</span> <span class="n">mmread</span>
<span class="n">array</span> <span class="o">=</span> <span class="n">mmread</span><span class="p">(</span><span class="s1">'soc-dolphins.mtx'</span><span class="p">)</span>
<span class="n">dolphins</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">from_scipy_sparse_array</span><span class="p">(</span><span class="n">array</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">dolphins</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">dolphins</span><span class="o">.</span><span class="n">edges</span><span class="p">())</span>
<span class="n">n</span><span class="p">,</span> <span class="n">m</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(62, 159)
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">unbiased</span> <span class="o">=</span> <span class="p">[</span><span class="n">dolphins</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">node</span><span class="p">)</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">dolphins</span><span class="p">]</span>
<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">unbiased</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([ 6, 8, 4, 3, 1, 4, 6, 5, 6, 7, 5, 1, 1, 8, 12, 7, 6,
9, 7, 4, 9, 6, 1, 3, 6, 3, 3, 5, 5, 9, 5, 1, 3, 10,
5, 1, 7, 11, 8, 2, 8, 5, 6, 7, 4, 11, 2, 6, 1, 2, 7,
10, 4, 2, 7, 2, 2, 9, 1, 5, 1, 3])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">empiricaldist</span> <span class="kn">import</span> <span class="n">Pmf</span>
<span class="n">pmf</span> <span class="o">=</span> <span class="n">Pmf</span><span class="o">.</span><span class="n">from_seq</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">pmf</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s2">""</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span>
<span class="n">xlabel</span><span class="o">=</span><span class="s2">"Friend count"</span><span class="p">,</span>
<span class="n">ylabel</span><span class="o">=</span><span class="s2">"Number of dolphins"</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s2">"Distribution of friend counts"</span><span class="p">,</span>
<span class="p">)</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">biased</span> <span class="o">=</span> <span class="n">resample_weighted</span><span class="p">(</span><span class="n">unbiased</span><span class="p">,</span> <span class="n">unbiased</span><span class="p">)</span>
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