Why Are Stock Prices Considered Log-normal?

Understanding key statistical concepts helps make sense of markets.

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Whether you invest in a single stock, a portfolio of shares or mutual funds made up of stocks, you should be familiar with the fundamental statistical concepts that can help explain the movements of stock prices. Normal and log-normal distributions are two key stat terms you will hear frequently. A basic understanding of these terms will help you grasp how stock prices evolve.

Normal Distribution

When a variable is normally distributed, its visual representation on a graph will have the familiar bell-curve shape. This means that moderate values have the highest probability of showing up, while extreme values have proportionally lower probabilities of materializing. Assume that bread sales on a typical Monday in a neighborhood grocery are normally distributed, with an average of 150 loaves. This means that sales levels that are significantly higher or lower than 150 loaves are proportionally less likely. Furthermore, a normal distribution is symmetrical, meaning that selling 150 - 20 (130 loaves) is equally likely as selling 150 + 20 (170 loaves).

Logartithm and Log-normality

While many variables around you may be normally distributed, a different type of pattern, known as log-normal distribution, may better describe the disbursement of other parameters. Log-normal simply means that the logarithm of the variable is normally distributed. The logarithm of a variable is the exponent to which another number must be raised to produce that number. While the math behind logarithms can get complex, suffice to say that logarithms are most appropriate if the variable at hand has a tendency to grow exponentially when its value is high but move little when its value is depressed.

Stock Prices

While the returns for stocks usually have a normal distribution, the stock price itself is often log-normally distributed. This is because extreme moves become less likely as the stock's price approaches zero. Cheap stocks, also known as penny stocks, exhibit few large moves and become stagnant. However, even the few small price changes you will see at these depressed levels correspond to large percentage changes, because the base is so low. For example, a 10-cent price change corresponds to a hefty 5 percent if the stock is only $2. So the stock's return is normally distributed, while the price movements are better explained with a log-normal distribution.


The distribution of stock prices and returns will help you determine the probable gains and losses in your portfolio. If most stocks in your portfolio have traditionally exhibited large moves on both the up and the down side, your potential losses as well as gains are large. Such a portfolio may be suited for a young investor, who has sufficient time until retirement to recover from a significant trading loss. An investor closer to retirement, on the other hand, may be better off with a portfolio that is less likely to gain or lose very much.