Normal Distribution

What Is the Normal Distribution?

The normal distribution is a pattern for how values are spread around a central point. When plotted on a graph, it forms a symmetrical, bell-shaped curve. The highest point sits at the mean, which is the average, and the curve falls away evenly on both sides. Because the shape is symmetric, values above and below the average are equally likely.

Two numbers define the curve: the mean and the standard deviation, which measures how widely values spread out. A small standard deviation produces a tall, narrow bell, while a large one produces a flatter, wider curve.

How It Works

In a normal distribution, the spread follows a predictable pattern. Roughly 68% of values fall within one standard deviation of the mean, about 95% fall within two, and about 99.7% fall within three. This is often called the empirical rule.

Many natural measurements, such as heights in a large group, approximately follow this shape. The central limit theorem helps explain why: averages of many independent random values tend toward a normal pattern even when the underlying values do not.

A Simple Example

Suppose a hypothetical investment has monthly returns that average 1% with a standard deviation of 4%. If those returns were normally distributed, about 68% of months would fall between -3% and +5%, and about 95% would fall between -7% and +9%. The numbers show how the standard deviation sets the typical range of outcomes.

Use in Investing

Investors encounter the normal distribution mainly through volatility and risk models. Some widely used models assume that daily stock returns follow a normal distribution, which makes calculations simpler. For example, a Value at Risk estimate built on this assumption uses the standard deviation of past returns to estimate how large a loss might be over a set period.

The assumption has limits. Real market returns often show fat tails, meaning extreme moves happen more frequently than a normal curve predicts. Returns can also be skewness|skewed rather than symmetric. Models that rely too heavily on normality can therefore understate the chance of large losses. Understanding this gap helps investors read risk figures with appropriate caution.