Nature's design can seem random upon first glance; however, a closer look reveals a common pattern that ties it all together, from the growth of a tree bark to the toss of a coin -- the bell curve. This symmetrical curve shows data points clustered around a mean, and progressively few data points further from the mean -- forming the "tails" of the curve.

Perhaps you first came across this curve in your statistics textbook explaining the normal distribution, or a biological chart of heights and weights. It turns out that the bell curve appears all around us in our day-to-day lives, from a microscopic level, like measuring molecular energies, to a macroscopic level, such as the distribution of SAT test scores.

It's no coincidence that all of these phenomena can be described by this smooth curve. There are some commonalities amongst these observations. First of all, they all have a "mean" that is represented by the peak of the curve. Characteristics like human height and weight are not uniformly distributed but rather trend toward a "normal" height, which is more probable and away from "extreme" heights that are less probable. This fact is crucial to the principle of natural selection, allowing this mean point to be the designated characteristic for the fittest individual.

Also, variance in these variables gives the bell curve its width. If humans all had the same height, a plot of their height distribution would be a rather boring vertical line at, say, 2 feet. In reality, we have varying heights, giving the bell curve its rounded shape. Similarly, in a classroom full of students, each one is unique with certain height, weight, IQ and perhaps a unique midterm score. These variations are a natural part of life and reflect the diversity present in any population.

Looking closer at midterm scores, we see just how many factors contribute to variance. Some of the variation can be explained by predictable factors, such as differences in intelligence or time spent studying. These factors can be broken down even further by considering course load or genetic influences on intelligence. However, not all factors are predictable. A student may have been feeling unwell on the exam day or struggled to focus -- these random and uncontrollable events add further to the variance in exam scores. Despite the complexity of these individual differences, it is precisely this variance that allows the scores of the whole class to form a neat, predictable curve.

Variables like height are continuous, meaning that their values change smoothly without jumping from one to the next. For example, a population of individuals have heights of 1.65 meters, 1.66 meters and every value in between. It seems natural for these continuous variables to form a smooth curve. In contrast, discrete variables are limited to take only fixed values within a range: If you rolled a coin 10 times and counted the number of heads you get, you could get three or four heads but not 3.5 heads. Shoe size is also discrete, increasing in increments of 0.5 -- it is impossible to have a shoe size of 9.25. So, how can the distributions of these discrete variables map out a continuous bell curve?

Imagine if you tossed a coin over and over again. On the first toss, you have a 0.5 probability of getting heads or tails. On the second toss, 0.25 of obtaining two heads, two tails and 0.5 for one head and one tail. As you keep tossing, probabilities for each outcome neatly align themselves in a smooth bell curve. This phenomenon is an illustration of the central limit theorem in mathematics and is key to why variables with a mean and positive variance eventually can be approximated to a bell curve.

A consequence of these measurements' tendency to a bell curve is used in a wide variety of disciplines. Investors may use a bell curve to model outcomes for predicting future behavior of stock prices. Scientists use a bell curve to model many phenomena, such as the Brownian motion of atoms in galaxies and the growth of many species.

While nature behaves discretely and randomly on a small, microscopic scale, it is on a larger scale that we observe nature freely. A gas atom may move in random, discrete steps, but when taking the movement of all the atoms together, we see a continuous distribution. Observables in nature are abundant with randomness, but on a larger scale, an elegant and rather simple looking curve emerges.