It was precisely 200 years ago that child prodigy Johann Carl Friedrich Gauss discovered the power of the normal distribution as a probability density function (PDF). He was interested in how to best approximate the value of an unknown quantity based on a series of measurements that have been taken with varying error. Gauss starts by following the common knowledge which implies that the most reasonable approximation should result in the average of the measurements, and hence ends up with the first version of the normal distribution as we know it today.
In the 200 years since, the normal distribution has become an indispensable element of statistical analysis. Whether they understand its application or not, pretty much everyone is familiar with some form of the bell curve. So whether you are a fellow admirer or yet to discover your appreciation for this beautiful phenomenon of nature, please indulge me with the privilege of sharing with you the 5 reasons why I love the normal distribution.
It’s Mathematically Elegant
The formula for the probability density function of the standard normal distribution is:
This formula is beautifully parsimonious, containing the two most well-known transcendental numbers π and e. For such an exceedingly powerful phenomenon, it is amazing that is reduces to such a short, elegant equation.
It’s Simple
The most striking feature of the normal distribution at first glance is its symmetrical, smooth bell shape, so let’s see why it looks this way. It is common to start by discussing distributions by looking at their two most fundamental properties: their central tendency and their spread. Central tendency is usually best indicated by the mean, μ, and the spread by the standard deviation, σ. For example, the standard normal distribution is defined as having μ = 0 and σ = 1. However, we may wish to use other measures to indicate central tendency, like median (the middle value of the sorted dataset) and the mode (the most commonly occurring value). Nonetheless, the normal distribution is particularly simple in that the mean, median and mode are all equal, which results in the bell curve being horizontally symmetrical. Additionally, the two points at one standard deviation either side of the mean are where the curve has its points of inflection, changing concavity from concave down to concave up. This gives the curve its bell-ish top and smoothens it out as deviation increases.
It’s Everywhere
Whether looking at the number of petals on each rose in a field or the weights of newborn babies, the normal distribution always appears to model data pretty accurately, The central limit theorem states that the larger the dataset, the more likely and accurately the normal distribution will approximate it. In fact, the application of this theorem is so remarkably widespread that, even in the case where a variable is not originally normally distributed, when samples of the dataset are taken, the distribution of the sample means will converge towards a normal distribution. There is nowhere you can look without finding the normal distribution hiding in plain sight.
It Helps Standardise Scores
It is often difficult to compare two scores from different domains. For instance, if we wish to compare the performance of a swimmer and a runner, it would require some analysis. Water is a more difficult medium to travel through than air, but it is hard to know just how much to adjust each race time to make them comparable. This is where the normal distribution, alongside the ever-applicable central limit theorem, can enable us to infer the answer by looking at how each sportsperson performed relative to their competitors, known as standardisation. By assuming that the set of swimming times and the set of running times are normally distributed, we can calculate the mean and standard deviation of each dataset. Now we can take each score, deduct its distribution’s mean and divide the result by its distribution’s standard deviation to obtain what is called a z-score. These z-scores are now directly comparable to each other, so if the swimmer has a lower z-score than the runner, it means that the swimmer’s time was in a lower percentile of its race than the runner’s, hence we may conclude that the swimmer performed better than the runner. Standardisation is thus a simple yet important way to compare scores from different contexts.
It Enables Predictive Analysis
One of the most powerful uses of the normal distribution is for predictive analysis. For example, businesses are often interested in predicting sales for the upcoming year to ensure they have enough inventory to meet demand, while also wanting to reduce the risk of incurring unnecessary inventory storage costs. In this case, sales would be considered the dependent variable, or outcome of interest. The business would then consider which variables that they expect will affect the outcome, or what the dependent variable depends on. These are called independent variables. Once the business has a dataset of these variables, it is able to take advantage of regression analysis to find a mathematical relationship that predicts the dependent variable in terms of the independent variables. A common technique is to use the ordinary least squares approach to determine a linear relationship, which is powerful enough on its own, however when the errors of the model are assumed to be normally distributed, then it is possible to evaluate the effectiveness of the model – and a model is only as good as its justification. The normal distribution demonstrates yet again how helpful it is in so many different areas.
Hopefully by now you are convinced of how magical and legendary the normal distribution is. Hold your respect for this phenomenon high and it shall come back to reward you. Happy analysing!
Cover image by Elias Tsolis from Pixabay