Understanding Normal Distribution: Key Concepts and Financial Uses

What Is a Normal Distribution?

Normal distribution, often referred to as the Gaussian distribution, is fundamental to statistics and finance due to its symmetry around the mean, forming the characteristic "bell curve." This distribution is critical in various financial analyses and decisions where most data points tend to cluster around an average value. The normal distribution is not only the foundation of many statistical theories, such as the Central Limit Theorem, but it is also key in assessing financial market behaviors, helping investors and analysts determine if financial instruments are priced fairly or deviate from expected norms. This article explores the properties, uses, and limitations of normal distributions to aid in informed financial decision-making.

Key Properties of Normal Distribution Explained

In technical stock market analysis, the normal distribution is commonly assumed. It has two parameters: mean and standard deviation. Here, mean (average), median (midpoint), and mode (most frequent observation) are equal, creating a symmetrical bell curve centered around the mean.

The normal distribution model is central to the Central Limit Theorem (CLT), which states that the averages of independent, identically distributed variables usually follow a normal distribution, regardless of the original distribution.

The normal distribution is one type of symmetrical distribution. Symmetrical distributions occur when a dividing line produces two mirror images. Not all symmetrical distributions are normal since some data could appear as two humps or a series of hills in addition to the bell curve that indicates a normal distribution.

Understanding the Empirical Rule in Normal Distribution

In normal distributions, 68.2% of data falls within one standard deviation of the mean, 95.4% within two, and 99.7% within three.

This fact is sometimes called the "empirical rule," a heuristic that describes where most of the data in a normal distribution will appear. Data falling outside three standard deviations ("3-sigma") would signify rare occurrences.

Analyzing Skewness in Normal Distribution

Skewness measures the degree of symmetry of a distribution. The normal distribution is symmetric and has a skewness of zero. If the distribution of a data set instead has a skewness less than zero, or negative skewness (left-skewness), then the left tail of the distribution is longer than the right tail; positive skewness (right-skewness) implies that the right tail of the distribution is longer than the left.

Exploring Kurtosis in Normal Distribution

Kurtosis measures the thickness of the tail ends of a distribution to the tails of a distribution. The normal distribution has a kurtosis equal to 3.0. Distributions with larger kurtosis greater than 3.0 exhibit tail data exceeding the tails of the normal distribution (e.g., five or more standard deviations from the mean).

This excess kurtosis is known in statistics as leptokurtic, but is more colloquially known as "fat tails." The occurrence of fat tails in financial markets describes what is known as tail risk. Distributions with low kurtosis less than 3.0 (platykurtic) exhibit tails that are generally less extreme ("skinnier") than the tails of the normal distribution.

The Formula for Normal Distribution

The normal distribution follows the following formula. Note that only the values of the mean (μ ) and standard deviation (σ) are necessary

where:

• x = value of the variable or data being examined and f(x) the probability function
• μ = the mean
• σ = the standard deviation

Financial Applications of Normal Distribution

The assumption of a normal distribution is applied to asset prices and price action. Traders may plot price points to fit recent price action into a normal distribution. The further price action moves from the mean, in this case, the greater the likelihood that an asset is being over or undervalued. Traders can use the standard deviations to suggest potential trades. This type of trading is generally done on very short time frames as larger timescales make it much harder to pick entry and exit points.

Many statistical theories model asset prices by assuming normal distribution. However, actual price data often show fat tails, with more frequent extreme movements, leading to kurtosis above three. Even if past data fit a normal distribution, it doesn't ensure future performance will.

Real-World Example of Normal Distribution

Many naturally occurring phenomena appear to be normally distributed. For example, the average height of a human is roughly 175 cm (5' 9"), counting both males and females.

As the chart below shows, most people conform to that average. Taller and shorter people exist with decreasing frequency in the population. According to the empirical rule, 99.7% of all people will fall with +/- three standard deviations of the mean, or between 154 cm (5' 0") and 196 cm (6' 5"). Those taller and shorter than this would be rare (just 0.15% of the population each).

The Bottom Line

Normal distribution, also known as the Gaussian distribution, is a probability distribution that appears as a "bell curve" when graphed. The normal distribution describes a symmetrical plot of data around its mean value, where the width of the curve is defined by the standard deviation.