# Normal Distribution

> The normal distribution is the bell curve behind many financial models. Learn its properties, the 68 95 99.7 rule, where traders use it and why markets break it.

Source: https://learn.tradelabsai.com/math/normal-distribution/  
Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Normal Distribution", https://learn.tradelabsai.com/math/normal-distribution/

The normal distribution, also called the Gaussian distribution or bell curve, is the most widely used distribution in statistics and finance. It is symmetric around its mean and fully described by two numbers: the mean and the standard deviation. Many financial models, from Markowitz portfolio theory to parts of the Black Scholes model and parametric value at risk, assume normally distributed returns. It is a useful approximation, but markets regularly produce extreme moves that the normal distribution says should almost never happen.

## Properties

| Property | Detail |
|---|---|
| Shape | Symmetric bell curve |
| Parameters | Mean (μ) and standard deviation (σ) |
| Mean, median, mode | All equal |
| Skewness | 0 |
| Kurtosis | 3 (excess kurtosis 0) |
| Tails | Thin: extreme values are very rare |

```
f(x) = (1 / (σ √(2π))) × e^(-(x - μ)² / (2σ²))
```

## The 68 95 99.7 rule

| Range | Probability inside | Probability outside (both tails) |
|---|---|---|
| μ ± 1σ | 68.3% | 31.7% |
| μ ± 2σ | 95.4% | 4.6% |
| μ ± 3σ | 99.7% | 0.27% |
| μ ± 4σ | 99.994% | About 0.006% |
| μ ± 5σ | 99.99994% | About 0.00006% |

## Where traders use it

| Use | Lesson |
|---|---|
| Converting volatility into expected ranges | [Implied Volatility (IV)](https://learn.tradelabsai.com/volatility/implied-volatility/) |
| Z scores and standardised signals | [Percentiles, Quantiles and Z-Scores](https://learn.tradelabsai.com/math/z-scores/) |
| Parametric value at risk | [Value at Risk (VaR)](https://learn.tradelabsai.com/portfolio/value-at-risk/) |
| Confidence intervals for averages (via the central limit theorem) | [Central Limit Theorem](https://learn.tradelabsai.com/math/central-limit-theorem/) |
| Black Scholes uses normal log returns | [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/) |
| Mean variance portfolio optimisation | [Modern Portfolio Theory and the Efficient Frontier](https://learn.tradelabsai.com/portfolio/modern-portfolio-theory/) |

**Example: Expected range from volatility**
A stock trades at $100 with annual volatility of 30%. Over one month, the standard deviation of returns is about 30% / √12 ≈ 8.7%. Under a normal approximation, there is about a 68% chance the stock ends the month between roughly $91 and $109, and about a 95% chance between roughly $83 and $117. These ranges help set stops, targets and option strikes, as long as you remember that real moves beyond them happen more often. See [Historical and Realized Volatility](https://learn.tradelabsai.com/volatility/historical-volatility/).

## Why markets are not normal

| Feature | Reality vs normal assumption |
|---|---|
| Fat tails | Extreme moves happen far more often. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/) |
| Skewness | Equity indices tend to have large down moves more often than large up moves |
| Volatility clustering | Calm and turbulent periods cluster, rather than each day being independent. See [GARCH](https://learn.tradelabsai.com/math/garch/) |
| Jumps | Prices gap on news |

**Example: Black Monday**
On 19 October 1987, the Dow Jones Industrial Average fell about 22.6% in a single day. Measured against the volatility of the preceding period, that was a move of more than 20 standard deviations. Under a normal distribution, such an event should essentially never occur in the lifetime of the universe. It did. See [Black Monday 1987](https://learn.tradelabsai.com/history/black-monday-1987/).

## Living with the normal distribution

- **Use it as a first approximation,** not as truth.
- **Stress test** beyond normal ranges. See [Stress Testing and Scenario Analysis](https://learn.tradelabsai.com/portfolio/stress-testing/).
- **Use fat tailed alternatives** such as the Student's t distribution for risk. See [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/).
- **Size positions** assuming extreme moves will happen. See [Position Sizing](https://learn.tradelabsai.com/risk/position-sizing/).
- **Look at empirical data** rather than relying only on formulas.

## Normal vs lognormal

Prices cannot fall below zero, but a normal distribution allows negative values. Models therefore often assume log returns are normal, which makes prices lognormal. See [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/).

## Testing for normality

The Jarque Bera test checks whether skewness and kurtosis match a normal distribution; daily stock and index returns almost always fail it. A Q Q plot, which compares the data's quantiles with normal quantiles, shows where the mismatch is: usually the points bend away at both ends, revealing fat tails.

## Frequently asked questions

### What is the normal distribution?

A symmetric, bell shaped probability distribution defined by its mean and standard deviation, with most values clustered near the mean and thin tails.

### What is the 68 95 99.7 rule?

About 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three, for a normal distribution.

### Why is the normal distribution dangerous for risk management?

Because it greatly underestimates the frequency of extreme moves, which happen much more often in real markets.

Next, learn the distribution used for prices in [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/).

## Continue learning

- Next lesson: [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/)
- Previous lesson: [Probability Distributions Explained](https://learn.tradelabsai.com/math/probability-distributions/)
- Related: [Probability Distributions Explained](https://learn.tradelabsai.com/math/probability-distributions/): Probability distributions describe the range and likelihood of outcomes. Learn the main ones used in trading, their shapes and when each applies.
- Related: [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/): Fat tails mean extreme market moves happen far more often than the normal curve predicts. Learn the evidence, the causes, how to measure them and how to manage them.
- Related: [Percentiles, Quantiles and Z-Scores](https://learn.tradelabsai.com/math/z-scores/): A z score shows how many standard deviations a value is from its mean. Learn the formula, its uses in mean reversion and pairs trading, and the pitfalls.
- Related: [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/): The Student's t distribution has fatter tails than the normal. Learn how it is used for small sample tests and to model fat tailed returns, with examples.
- Related: [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/): The Black Scholes model prices European options from five inputs. Learn the formula, its assumptions, a step by step example and where the model breaks down.
- Related: [Value at Risk (VaR)](https://learn.tradelabsai.com/portfolio/value-at-risk/): Value at risk estimates the loss a portfolio should not exceed with a given confidence over a set period. Learn the three methods, an example and the limits.