Realizing Gaussian law of randomness

July 12, 2016 Maths Musings

During our school / college days, we would have studied random distributions, probability, mean etc. We would have also studied the famous Gauss Law or more famously known for Bell Curve.

In simple words, Gauss Law predicts that ( it proved way back though!!) any randomness has a pattern and the pattern looks like a bell curve.

$Image of BellCurve$

This act of randomness is true irrespective of the values / domains on which this randomness prevail. Hence this bell curve distribution has been used as a measurement tool in variety of domains like errors / noises and even in corporate appraisal cycles to rank employees.

Though this information is well known / well tutored right from our class X, I didn’t realize the truth behind this, until I saw a Ted video about Mathematics from Cédric Villani, where he explains about the applications of mathematics and it’s pure abstract nature.

In the talk, he demonstrated how Gauss Law a.k.a Law of errors can be applied to anything, by using a simple device which transports tiny beads from top to bottom via various random paths. Each bead take a random path to reach the bottom and that pattern always follow this bell curve.

My experiment to realize the truth

Hello World

Quite intrigued by this video, I set out to test the randomness pattern myself. First experiment was to play with the rand() function which is available in almost every language. The intent was to generate a number randomly ( not a random prime number ) for a given period and then plot them to see what pattern it is taking.

<html>
  <head>
    <script type="text/javascript" src="https://www.gstatic.com/charts/loader.js"></script>
    <script type="text/javascript">
      google.charts.load('current', {'packages':['corechart']});
      google.charts.setOnLoadCallback(drawChart);

	  var gdata;
	  
	  function getRandomArbitrary(min, max) {
		return Math.floor(Math.random() * (max - min) + min);
		}
		
      function drawChart() {
        var data = google.visualization.arrayToDataTable([
          ['Year', 'Sales', 'Expenses'],
          ['2004',  getRandomArbitrary(1,1000), getRandomArbitrary(1,1000)],
          ['2005',  getRandomArbitrary(1,1000), getRandomArbitrary(1,1000)],
          ['2006',  getRandomArbitrary(1,1000), getRandomArbitrary(1,1000)],
          ['2007',  getRandomArbitrary(1,1000), getRandomArbitrary(1,1000)],
		  ['2008', getRandomArbitrary(1,1000), getRandomArbitrary(1,1000)]
        ]);
		

        var options = {
          title: 'Ramdom distribution experiment',
          curveType: 'function',
          legend: { position: 'bottom' }
        };

        var chart = new google.visualization.LineChart(document.getElementById('curve_chart'));

        chart.draw(data, options);
		});
      }
    </script>
  </head>
  <body>
    <div id="curve_chart" style="width: 900px; height: 500px"></div>
  </body>
</html>

You can see a live preview of the chart, refreshed at regular intervals below. As you can see, the randomness in this experiment, do always follow the Gauss Law

Actual World

As you can probably guess, above experiment was very brute-force. The rand() has been engineered to follow the bell curve i.e normal distribution/ Hence there is no surprise in the output. It did not really help to prove the fact that, Gauss Law is universal.

However, using this code as base, I tried to get the World population of countries in random way. Luckily there exists a API available to retrieve the information about every country in the world. Using this, I tried to query the population of a random country and tried to plot that in the graph.

Remember this act is truely random i.e I never know the population of each country before in hand + the choice of the country is not pre-decided. Everything changes at the start of every sample of experiment.

Below iFrame would show you the live preview of the graph. As you can see, the randomness in this experiment, do always follow the Gauss Law.

Math is Natural Science

This once property of uniform abstraction in mathematics, really amazes me. As rightly put by Dr Cédric Villani in his talk,

“Mathematics allows us to go beyond the intuition and explore territories which do not fit within our grasp.”.

Cya :-)