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Key Takeaways
- A normal distribution is fully described by two numbers: its mean, which sets the center, and its standard deviation, which sets the width.
- The empirical rule states that about 68% of outcomes lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
- A normal model converts standardized distances into probabilities, so a daily volatility gives daily ranges and an annual volatility gives annual ones.
- A z-score measures distance from a mean. A distribution supplies the corresponding probability.
- For option pricing, distinguish normal log returns from lognormal terminal prices, and terminal outcomes from path-dependent events.
What is a normal distribution?
A normal distribution is a symmetric, bell-shaped probability model described by two numbers: its mean, which sets the center, and its positive standard deviation, which sets the width. Mean, median and mode coincide at the peak. Traders use it to turn a mean return and a volatility estimate into ranges and probabilities.
The total area under the density is one. The area between two values gives the probability of an outcome in that interval. An individual exact value has zero probability in this continuous distribution.
How are normal distribution probabilities calculated?
Subtract the mean from an outcome and divide by standard deviation to obtain its z-score. For a normal variable, that standardized result follows the standard normal distribution with mean zero and standard deviation one.
The cumulative function Phi(z) gives the probability below z. A range probability subtracts the cumulative probability at its lower boundary from the cumulative probability at its upper boundary.
Formula
z = (x − μ) / σ; f(x) = exp(−z² / 2) / (σ × √(2 × pi)); P(a ≤ X ≤ b) = Φ((b − μ) / σ) − Φ((a − μ) / σ)- x is an observed outcome, such as a daily or annual return.
- The mean (μ) is the average outcome and locates the peak of the curve.
- The standard deviation (σ) measures dispersion and sets how wide the bell is.
- σ is positive. Phi is the standard normal cumulative distribution function; z expresses distance from the mean in standard deviations.
Normal distribution example: the 68-95-99.7 rule
A trader models a stock’s annual return with a normal distribution using a mean of 8% and a standard deviation of 15%. The trader wants the ranges that the empirical rule implies.
- One standard deviation band: 8% − 15% = −7% low, 8% + 15% = 23% high
- About 68% of annual outcomes fall between −7% and +23%
- Two standard deviation band: 8% +/- 30% spans −22% to +38%, covering about 95%
- Three standard deviations: 8% +/- 45% spans −37% to +53%, covering about 99.7%
The model assigns 68.27% to −7% through 23%, 95.45% to −22% through 38%, and 99.73% to −37% through 53%. Outside the two-sigma interval is 4.55% total probability, split equally between its two tails.
Set mean return to 8% and standard deviation to 15 percentage points. The one-sigma interval is −7% to 23% and contains 68.27% of the model’s probability.
Two sigma gives −22% to 38% and contains 95.45%. Three sigma gives −37% to 53% and contains 99.73%.
The total probability outside the two-sigma interval is 4.55%. A year worse than −22% is one of those two tails, so its chance is half that: 2.275%.
Which calculations require a normal distribution?
A z-score can standardize observations from many distributions. The normal cumulative function supplies normal-model probabilities specifically.
Likewise, variance, standard deviation and the Sharpe ratio can be calculated without assuming normality. Normality is what lets you read them as probabilities, such as a 68.27% chance of finishing within one standard deviation of the mean.
Normal vs lognormal models and the time horizon
A normal price model describes additive price changes. A lognormal price model describes prices whose logarithms are normal. Black-Scholes uses the second construction.
Square-root-of-time scaling follows from adding uncorrelated, equal-variance increments, so check that returns behave that way before you stretch a daily volatility to a month.
A terminal distribution answers where the outcome finishes. A question about touching a price or remaining within a band throughout a period also requires a model of the path. Read an options position’s payoff at the one- and two-sigma prices in the options profit calculator, then shift volatility to see how a wider distribution moves those outcomes.
Limitations of the normal distribution
- Check tail probabilities against the asset’s own return history. Market returns often show fatter tails than the bell curve: daily S&P 500 returns have clustered near the mean more tightly than a normal model predicts, yet extreme days have occurred more often than it assigns, so normal tail probabilities can understate large moves.
- Simple stock returns cannot fall below −100%, while a normal variable extends without bound in both directions. Black-Scholes applies the normal curve to log returns, which keeps modeled prices above zero.
- Test the returns for skew, and take the mean and standard deviation from a look-back window that matches the period you are modeling. The curve is symmetric and fixed by those two estimates, so skewed returns or an unrepresentative window can produce misleading ranges.
- Square-root-of-time scaling assumes uncorrelated, equal-variance increments. A terminal distribution prices where the outcome finishes; a stop or barrier that price touches along the way needs a path model.
Bottom line
The normal distribution turns a mean and a standard deviation into ranges and probabilities, which gives traders a quick way to translate volatility into a plausible range of outcomes. Use inputs that match the variable and horizon, and check its tail estimates against the asset's actual history.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Fact File: S&P 500 Sigma Events
- Fidelity: Standard Deviation Indicator
- Federal Reserve: Risk, Uncertainty, and Expected Returns (FEDS 2011-45)
How entries are sourced, calculated and updated: editorial methodology.