What you will learn
- Recognize the shape and features of the normal distribution
- Understand how the mean and standard deviation define a normal curve
- Apply the 68-95-99.7 empirical rule
- Know why the normal distribution matters in finance and where it falls short
One distribution appears again and again across nature, science, and finance: the normal distribution, better known as the bell curve. Heights, measurement errors, test scores, and many financial quantities cluster into this familiar shape. Understanding it opens up a lot of practical statistics, because many methods assume data is roughly normal. This lesson builds your intuition for the shape and then hands you a rule of thumb you will use often.
The normal distribution, clearly explained (StatQuest)
A friendly introduction to the bell curve and how its mean and standard deviation control its center and width.
The shape of the bell curve
A normal distribution has a distinctive shape. It is symmetric, meaning the left and right halves mirror each other. It has a single peak in the middle, right at the mean, where values are most common. And it tapers off smoothly on both sides, so values far from the mean get rarer and rarer but never quite reach zero. Two numbers completely describe any normal distribution. The mean sets where the center sits, and the standard deviation sets how wide and spread out the curve is. A small standard deviation gives a tall, narrow bell. A large one gives a short, wide bell. Because it is symmetric, the mean, median, and mode of a normal distribution all sit at the same point in the center.
The 68-95-99.7 rule
Here is the most useful fact about the normal distribution, and it is worth memorizing. For any normal distribution, no matter its mean or standard deviation, a fixed fraction of the data falls within a given number of standard deviations of the mean. About 68 percent of all values fall within one standard deviation of the mean. About 95 percent fall within two standard deviations. And about 99.7 percent fall within three standard deviations. This is called the empirical rule, and it lets you turn a mean and a standard deviation into a quick picture of where the data lives and how unusual any given value is.
| Range around the mean | Share of the data |
|---|---|
| Within 1 standard deviation | About 68 percent |
| Within 2 standard deviations | About 95 percent |
| Within 3 standard deviations | About 99.7 percent |
| Beyond 3 standard deviations | About 0.3 percent (very rare) |
The empirical rule (Khan Academy)
Works through the 68-95-99.7 rule with a concrete example. Watch how each range around the mean captures its share of the data.
Applying the empirical rule to returns
Suppose a stock's annual return is roughly normal, with a mean of 8 percent and a standard deviation of 15 percent. Between what two returns do about 68 percent of years fall?
- Find one standard deviation below the mean. 8 percent minus 15 percent is negative 7 percent.
- Find one standard deviation above the mean. 8 percent plus 15 percent is 23 percent.
- Read the range. About 68 percent of years fall between negative 7 percent and positive 23 percent.
Why it matters: The mean and standard deviation, plus the empirical rule, instantly describe the typical range of outcomes. This is why standard deviation is used as the standard measure of risk.
Use the empirical rule
A fund's annual return is roughly normal with a mean of 10 percent and a standard deviation of 5 percent. About 95 percent of years fall within two standard deviations. What is the UPPER end of that 95 percent range, in percent?
Why it matters, and where it fails
The normal distribution is central to finance. Many models of returns, risk, and option pricing assume returns follow a roughly normal shape, and standard deviation used as risk only has its clean 68-95-99.7 meaning because of this assumption. But here is the caution, and it is one the previous unit hinted at. Real financial returns are not perfectly normal. They have fat tails, meaning extreme events like market crashes happen far more often than a normal distribution predicts. A true normal distribution says a five or six standard deviation daily move should essentially never occur, yet markets produce them every so often. Assuming normality can therefore badly underestimate the risk of rare disasters, which is how some famous blowups happened. Use the normal distribution as a useful approximation, but never forget the tails are fatter than it claims.
The bell curve is the most useful bad assumption in finance. It describes the ordinary days well and understates the catastrophes that actually matter.
Spot the flawed assumption
A risk model assumes daily stock returns are perfectly normal and concludes that a 20 percent one-day crash is so many standard deviations away it should never happen in the life of the universe. Yet such crashes have occurred. What went wrong?
The model assumed normality, but real returns have fat tails. Extreme moves happen far more often than a normal distribution predicts, so the model badly underestimated crash risk.Describe the empirical rule
In your own words, explain what the 68-95-99.7 rule tells you and why it is useful when you only know a mean and a standard deviation.
Write an answer before comparing it with the model response.
Model answer
The rule says that for any normal distribution, about 68 percent of the data falls within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. It is useful because if I only know the mean and standard deviation of something roughly normal, I can immediately picture the range of typical outcomes and judge how unusual any particular value is. For example, a value more than two standard deviations from the mean is in the rarest 5 percent, so it stands out as unusual.
The empirical rule works by counting standard deviations from the mean. The next lesson gives that idea a precise name and formula: the z-score, which measures exactly how many standard deviations any value sits from the mean.