Normal Distribution

Normal Distribution

The normal distribution, or Gaussian distribution, is a foundational probability distribution in statistics, appearing as the familiar “bell curve.” It is symmetric about its mean and widely used to model natural phenomena and underpin statistical theory, including the Central Limit Theorem.

Key takeaways

• Symmetric, bell-shaped distribution centered at the mean; mean = median = mode.
• Fully described by two parameters: mean (μ) and standard deviation (σ).
• Empirical rule: ~68.2% of data within ±1σ, ~95.4% within ±2σ, ~99.7% within ±3σ.
• Skewness = 0 and kurtosis = 3 for a true normal distribution. Deviations (skew or fat tails) matter in practice.
• Common in theory and many applications, but financial data often show skewness and fat tails that violate the normal assumption.

Properties

• Parameters: mean (μ) locates the center; standard deviation (σ) controls spread.
• Symmetry: left and right sides are mirror images; mean = median = mode.
• Central Limit Theorem: averages of many independent, identically distributed variables tend toward a normal distribution, regardless of the original distribution (under broad conditions).
• Standard normal: the special case with μ = 0 and σ = 1; values are often converted to z-scores: z = (x − μ) / σ.

Empirical rule

For a normal distribution:
* ≈68.2% of observations lie within ±1σ of the mean.
≈95.4% within ±2σ.
≈99.7% within ±3σ.
Observations beyond ±3σ are rare under normality and are often treated as outliers.

Skewness

Skewness measures asymmetry:
* Normal distribution has skewness = 0.
* Negative skew (left-skew) means a longer left tail; positive skew (right-skew) means a longer right tail.
Skewness indicates potential bias away from symmetry and affects tail risk and summary measures.

Kurtosis

Kurtosis measures tail heaviness:
* Normal distribution has kurtosis = 3.
* Excess kurtosis = kurtosis − 3.
* Positive excess kurtosis (leptokurtic) → fat tails, more extreme events than normal.
* Negative excess kurtosis (platykurtic) → thinner tails.
In finance, fat tails are important because they imply more frequent large losses or gains than predicted by a normal model.

Formula

The probability density function (PDF) of a normal distribution is:

f(x) = (1 / (σ * sqrt(2π))) * exp(−(x − μ)² / (2σ²))

where x is the value, μ is the mean, σ is the standard deviation, and exp denotes the exponential function.

Uses in finance

• Risk and return models often assume normality for convenience (e.g., portfolio theory, option pricing approximations, VaR calculations).
• Traders may use standard deviations from recent means to identify potential over- or under-valued prices on short time frames.
• Caution: financial returns frequently show skewness and fat tails (higher kurtosis), so relying solely on normal assumptions can underestimate extreme risks.

Example

Human heights approximate a normal distribution: if the mean is 175 cm and σ ≈ 7 cm, then about 99.7% of people fall between 175 ± 3·7 → roughly 154 cm to 196 cm. Values outside this range are rare under normality.

Limitations

• Many real-world data sets—especially financial returns—deviate from normality (skewness, kurtosis, volatility clustering).
• Alternatives (log-normal, Student’s t, stable distributions, nonparametric methods) may better capture tail behavior, but choosing and estimating alternatives introduces complexity.

Bottom line

The normal distribution is a useful, tractable model for symmetric, unimodal data and underlies many statistical methods. However, practitioners should test the normality assumption and be mindful of skewness and fat tails, particularly in finance where extreme outcomes are more common than the normal model predicts.