From Correlation to Causation through stories and math

Correlation and causation are two concepts that people often mixup in their minds. I must admit that I myself have been guilty about this, and it unlikely that I would ever entirely grow out of it as it is wired deeply into our psychology. Let me use this article to briefly emphasise what the concepts of correlation and causation means, some interesting stories that have emerged from people misunderstanding these concepts and an algorithm that attempts to find causal relationship using correlation information. Here is a story that I heard a professor of mine, Prof. Dr. Ernst-Jan Camiel Wit, tell us during a lecture. There was a school that was involved in a study to see if providing free mid-day meals to students, which they could choose to be subscribed to this or not. At the end of the study, both the students who subscribed to it and did not where tested for different health indicators. It was observed that the students who chose to have meals from the programme had poorer health

Estimating the value of Pi using Monte Carlo Method

The very first moment I was shown this, I was amazed. Estimating something using random events is pretty amazing. The first time I did this in MATLAB, but now, I think I will attempt to do the same in JavaScript.

The undelying concept of Monte Carlo Method is to use randomness to determine the value. The method is often used to solve problems that are hard to determine exactly, when there is enough statistical data available.

Let us define a square centered at the origin with side of length 2. A circle centered at the origin is inscribed in this square.

The assumption that we are using in this method is that if we randomly take a few points within the square, the ratio of the number of points that falls within the circle to the total points taken is equal to that of the area of the square and the circle. That is,

Area of square / area of circle = totalRandomSamples / totalPointsInCircle
Let us write a JavaScript program that could do perform this with a huge number of random sample points generated on the go. To check if the generated point lies within the circle or not, we use pythogoras theorem to calculate the distance from the center.

var squareSide = 2;
var circleRadius = squareSide/2; 
var totalRandomSamples = 100000;
var totalPointsInCircle = 0; 
for(var i = 0; i < totalRandomSamples; i++){
    // A random point contained within the square
    var newX = Math.random() * squareSide - squareSide/2;
    var newY = Math.random() * squareSide - squareSide/2;
    // Check if the new coordinate is inside or outside the circle
    if(Math.sqrt(newX * newX + newY * newY) < circleRadius){
// Area of square / area of circle = totalRandomSamples / totalPointsInCircle
// (squareSide * squareSide) / (pi * circleRadius * circleRadius) = totalRandomSamples / totalPointsInCircle
var pi = squareSide * squareSide * totalPointsInCircle / (circleRadius * circleRadius * totalRandomSamples); 

The value of 𝜋 calculated using the method may vary everytime one runs the code. If you wish to improve the accurace, you can increase the number of Random Samples being used. Also, a better algorithm for generating random points may also be used.

I find that the value calculated is usually within 1% accuracy. The fact that this method simply uses cartesian coordinates to calculate 𝜋 is something that blew my mind.


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