What Is Normal Distribution?

What is the normal distribution?

When people think of a bell curve, they're picturing the distribution that's known as a "normal distribution." Under a normal distribution, the bulk of the values of a set of numbers cluster around the middle value, and gradually get less concentrated as you move to the ends of the chart.

This may seem like a small piece of information, but it's crucial for understanding how basic statistics work and is the basis for a ton of financial and statistical theories. The normal distribution is a model, and rarely plays out exactly the same in the real world, but it allows us to model the behavior of a set of data, such as the price-to-earnings (P/E) ratio distribution of a variety of stocks.

Key properties of normal distribution

Even though normal distributions are theoretical only, they represent a statistical concept that's invaluable. No matter where your normal distribution occurs, you'll find that it shares all of the following properties:

• Symmetry. No matter what you're looking at using a normal distribution, you'll find that the right and left halves are identical. This is part of what makes it a beautiful tool for understanding where numbers fall compared to others.
• Mean, median, and mode are equal. Because the model has perfect symmetry, the mean, median, and mode are always the same. This allows for a distinct midpoint without any confusion about which of these measures should be prioritized.
• Consistent proportions. There's a constant proportion of distance under the curve between the mean and any specific number of standard deviations from the mean, positive or negative, known as empirical rule. So, for example, 95% of cases may fall within two standard deviations from the mean in either direction.
• Skewness and kurtosis. The coefficients known as skewness and kurtosis describe how different any distribution is from the normal distribution. Skewness is a measure of symmetry, and kurtosis measures the thickness of the ends of the distribution, all in relation to the normal distribution.

Where is the normal distribution used?

The normal distribution is the basis of many important ways of evaluating your portfolio or individual stocks. Here are a few ways it's used in investing:

• Modeling stock returns. By assuming that stock returns follow a normal distribution, it's easier to predict the probability of different returns and the likelihood of a gain or loss.
• Risk management. Both value at risk (VaR) and volatility are based on the normal distribution.
• Pricing options. Models like Black-Scholes, which are used to determine how much an option should be worth today, are based on normal distributions.
• Benchmarking portfolios. Comparing your portfolio to a benchmark can help you estimate how well you're doing, both in luck and skill. This requires a normal distribution underneath the model to determine the probability of each.
• Modern Portfolio Theory (MPT). This portfolio construction theory is used to build portfolios that are optimized for specific returns based on risk tolerance. It requires a normal distribution to help calculate correlations and covariances.

Why does normal distribution matter to investors?

Normal distribution is one of those things that you don't even know you're using a lot of the time. But whether you're looking at risk models or trying to figure out just how good of an investor you actually are, it's there, under the surface, supporting many different equations that you need to examine stocks and their performance.

Statistics and mathematical models are built on foundations like normal distribution, and without them, investors are flying completely blind, no matter how good they may be at picking stocks. You can't just be a "vibes" investor -- it's also important to look at the numbers and how they compare across a wide range of metrics. The normal distribution is one of those vital keys to unlocking the whole world of successful stock investing.