Normal Distribution: Definition, Properties, and Financial Applications

September 28, 2023 3 min read Economics Statistics Finance Normal Distribution Gaussian Distribution Statistics in Finance Empirical Rule Central Limit Theorem

Explore the concept of normal distribution or Gaussian distribution, its fundamental properties, how it's utilized in finance and its impact on statistical analysis and trading tactics.

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Introduction

Diving into the world of figures and probabilities, the normal distribution stands out as the celebrity of statistical data, impressing with its perfect symmetry and predictable behavior. Also known as the Gaussian distribution, after the prodigious mathematician Carl Friedrich Gauss, this probability distribution pattern is the backbone of endless levels of statistical analysis, particularly in the finance sector where Gaussian models assume most variables jog obediently on its well-behaved bell curve.

Properties of Normal Distribution

Behold the bell-shaped curve! This is where the magic happens:

Symmetry: The normal distribution is perfectly symmetrical around its mean, mirroring data points in a harmonious dance around the central value.
Mean and Standard Deviation: The mean, median, and mode of a normal distribution are like three musketeers, all located at the center of the curve. Here the mean is zero, and the standard deviation prowess dictates the curve’s width.
Empirical Rule: Often termed as the “68-95-99.7 Rule”, which, like well-rehearsed choreography, predicts that 68% of data within one standard deviation, 95% within two, and nearly all (99.7%) within three, make this rule a statistical superstar.

Skewness and Kurtosis

Not all distributions can join the normal group.

Skewness: If a distribution has skewness, it’s definitely gossiping behind the normal distribution’s back, lacking that perfect fold of symmetry.
Kurtosis: With a kurtosis grander than three, other distributions might show off heavier tails or sharper peaks, but normal distribution remains poised with a kurtosis of exactly three, neither fat-tailed nor peaky.

Formula

The mathematical harmony of the normal distribution is elegantly summed up with its formula:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$

Where $ \mu $ is the mean and $ \sigma $ the standard deviation.

Applications in Finance

Within the mazes of financial markets, assuming a normal distribution becomes a handy tool:

Risk Management: Financial wizards use the normal model to calculate probabilities of various outcomes based on historical data.
Trading Strategies: Traders often tweak their strategies based on how asset prices deviate from what Mr. Gaussian predicts.
Modeling and Forecasting: Despite its assumptions often not holding (cue the fat tails and outliers), it sets a base model from where adjustments can be plotted.

Central Limit Theorem: The superstar theorem in statistics supporting why means of samples tend to follow a normal distribution as sample size increases.
Standard Deviation: A measure of dispersion in a normal distribution, indicating how spread out the values are from their mean.
Z-Score: A statistical measurement describing a value’s relationship to the mean of a group of values, often used in standard normal distribution analysis.