If you hadn’t noticed, the icon of my website are five different normal distributions, each with different parameters. Let’s dive into the math first.

Mathematical description

The normal distribution is a continuous probability distribution. It’s sometimes called the “bell curve”. It’s ubiquitous in the real world. Many phenomena can be described by it. The value of the curve, for a given value $x$ is:

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{\frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^2}$$

This formula doesn’t tell you very much meaningful. For that, we need to integrate. If you’ve forgotten what integration means, for the purpose of this formula it’s simply the area under the curve. The following integral, let’s call it $p$:

$$p = \int^{0.5}_{-0.5} f(x)$$

tells you the area under the curve from $x=-0.5$ to $x=0.5$. So what does $p$, which is just a number (for fixed values of the parameters), represent? Well, it tells you the probability that the value of a random variable that is normally distributed falls between -0.5 and 0.5.

For example, if we take the parameters $\sigma=1$ and $\mu=0$ (a so-called “standard normal”, which is symmetric around zero and has a standard deviation of 1), we get $p \approx 0.38$. So the chance is around 38% that, if the variable is truly normally distributed, it falls somewhere between -0.5 and 0.5.

Why are normal distributions so common?

There are a lot of reasons for this. Take a look at this wonderful video by 3Blue1Brown if you want to learn more.

Choice for the icon

One of the most important things I think everyone should realize at all times, is just how inherently uncertain this incredibly complex world is. We cannot predict the future, many outcomes, good and bad, were just plain chance.

We have very limited control over much of the world, particularly over other people. That doesn’t mean you can’t do anything. No, I would state the opposite. Because this world is so uncertain, it’s our job to do our best to still influence it to make sure it’s a better place.

The normal distribution, arguably the most common probability distribution, shows we can try to account for a range of outcomes, but we’ll never know exactly what will happen, nor can we completely control it.