The Normal Distribution in Trading: Uses and Limits

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.

Advanced3 min readUpdated 3 Oct 2026

Lesson 23 of 46

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)
Z scores and standardised signals	Percentiles, Quantiles and Z-Scores
Parametric value at risk	Value at Risk (VaR)
Confidence intervals for averages (via the central limit theorem)	Central Limit Theorem
Black Scholes uses normal log returns	Black-Scholes Model
Mean variance portfolio optimisation	Modern Portfolio Theory and the Efficient Frontier

Why markets are not normal#

Feature	Reality vs normal assumption
Fat tails	Extreme moves happen far more often. See 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
Jumps	Prices gap on news

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.
Use fat tailed alternatives such as the Student's t distribution for risk. See Student's t-Distribution.
Size positions assuming extreme moves will happen. See 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.

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.

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Next lessonLognormal DistributionA lognormal distribution describes values whose log is normal, like prices that cannot go negative. Learn log returns, volatility drag and its option uses.