Calculating Pi with Skittles and Census Data - Numberphile

In this video, the presenters explore a creative and hands-on method for approximating the value of pi using random distribution. The experiment begins with a simple physical setup involving a circle, a square, and a large quantity of Skittles. To create the necessary geometric bounds, the presenters draw a circle and then use books—including a thesis on string theory and a volume titled *Fantastic Numbers*—to construct a square that perfectly encloses the circle. The core of the experiment relies on a statistical method. The presenters drop a handful of Skittles randomly over the area and then begin the tedious process of counting them. They first count the candies that landed inside the circle, arriving at a figure of 562. They then count those that landed outside the circle but still within the square, which totals 159. By adding these together, they establish a total population of 721 Skittles within the square. To find pi, they use the ratio of the Skittles inside the circle to the total number of Skittles in the square. Mathematically, they explain that if a circle has a radius of one, its area is pi. A square that perfectly fits that circle would have a side length of two, resulting in an area of four. Therefore, the ratio of the area of the circle to the area of the square is pi divided by four. By taking their experimental ratio (562 divided by 721) and multiplying it by four, they achieve an estimate of approximately 3.11. While not perfect, they consider it a respectable result given the "dodgy" Skittles sitting right on the line of the circle. The presenters then decide to scale this experiment up using real-world data rather than candy. They utilize population data from the UK’s Office of National Statistics to see if the distribution of people can provide a more accurate estimate of pi. The logic remains the same: they pick a center point, define a radius to create a circle, and then calculate the population within that circle versus the population within the bounding square. They test several locations across the UK with varying results. Starting in Nottingham with a 100-kilometer radius, they find a population of 16.2 million in the circle and 20.389 million in the square, yielding a pi estimate of 3.18. They then try Anfield in Liverpool, but the proximity to the coast and the "funny shape" of the coastline skew the results significantly, resulting in a poor estimate of 3.27. The experiment moves to Buckingham Palace in London with a smaller 10km radius. This attempt proves difficult because the population data is "spiky" and non-uniform in such a dense urban area, leading to an estimate of 3.28. Finally, they test the geographical center of England in Leicestershire. This location provides their best population-based result: an estimate of 3.049. Ultimately, the presenters conclude that the Skittles actually provided a better approximation than the population data. They attribute this to the fact that human beings do not distribute themselves randomly; we tend to "clump" in cities, and our geography is interrupted by oceans and irregular coastlines. The video ends by noting that while these physical and social experiments provide interesting approximations like 3.11 or 3.15, the true value of pi continues with much more precision, such as 3.14159.