# Normal Distribution: What It Is, Properties, Uses, and Formula

## What Is a Normal Distribution?

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

In graphical form, the normal distribution appears as a "bell curve".

### Key Takeaways

• The normal distribution is the proper term for a probability bell curve.
• In a normal distribution the mean is zero and the standard deviation is 1. It has zero skew and a kurtosis of 3.
• Normal distributions are symmetrical, but not all symmetrical distributions are normal.
• Many naturally-occurring phenomena tend to approximate the normal distribution.
• In finance, most pricing distributions are not, however, perfectly normal.

## Understanding Normal Distribution

The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses. The standard normal distribution has two parameters: the mean and the standard deviation.

The normal distribution model is important in statistics and is key to the Central Limit Theorem (CLT). This theory states that averages calculated from independent, identically distributed random variables have approximately normal distributions, regardless of the type of distribution from which the variables are sampled (provided it has finite variance).

The normal distribution is one type of symmetrical distribution. Symmetrical distributions occur when where 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.

## Properties of the Normal Distribution

The normal distribution has several key features and properties that define it.

First, its mean (average), median (midpoint), and mode (most frequent observation) are all equal to one another. Moreover, these values all represent the peak, or highest point, of the distribution. The distribution then falls symmetrically around the mean, the width of which is defined by the standard deviation.

All normal distributions can be described by just two parameters: the mean and the standard deviation.

### The Empirical Rule

For all normal distributions, 68.2% of the observations will appear within plus or minus one standard deviation of the mean; 95.4% of the observations will fall within +/- two standard deviations; and 99.7% within +/- three standard deviations. This fact is sometimes referred to as the "empirical rule," a heuristic that describes where most of the data in a normal distribution will appear.

This means that data falling outside of three standard deviations ("3-sigma") would signify rare occurrences.

### Skewness

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.

### Kurtosis

Kurtosis measures the thickness of the tail ends of a distribution in relation 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 the 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

## How Normal Distribution Is Used in Finance

The assumption of a normal distribution is applied to asset prices as well as price action. Traders may plot price points over time 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.

Similarly, many statistical theories attempt to model asset prices under the assumption that they follow a normal distribution. In reality, price distributions tend to have fat tails and, therefore, have kurtosis greater than three. Such assets have had price movements greater than three standard deviations beyond the mean more often than would be expected under the assumption of a normal distribution. Even if an asset has gone through a long period where it fits a normal distribution, there is no guarantee that the past performance truly informs the future prospects.

## Example of a Normal Distribution

Many naturally-occurring phenomena appear to be normally-distributed. Take, for example, the distribution of the heights of human beings. The average height is found to be roughly 175 cm (5' 9"), counting both males and females.

As the chart below shows, most people conform to that average. Meanwhile, taller and shorter people exist, but 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 quite rare (just 0.15% of the population each).

## What Is Meant By the Normal Distribution?

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. It is visually depicted as the "bell curve."

## Why Is the Normal Distribution Called "Normal?"

The normal distribution is technically known as the Gaussian distribution, however it took on the terminology "normal" following scientific publications in the 19th century showing that many natural phenomena appeared to "deviate normally" from the mean. This idea of "normal variability" was made popular as the "normal curve" by the naturalist Sir Francis Galton in his 1889 work, Natural Inheritance.

## What Are the Limitations of the Normal Distribution in Finance?

Although the normal distribution is an extremely important statistical concept, its applications in finance can be limited because financial phenomena—such as expected stock-market returns—do not fall neatly within a normal distribution. In fact, prices tend to follow more of a log-normal distribution that is right-skewed and with fatter tails. Therefore, relying too heavily on a bell curve when making predictions about these events can lead to unreliable results. Although most analysts are well aware of this limitation, it is relatively difficult to overcome this shortcoming because it is often unclear which statistical distribution to use as an alternative.

Article Sources
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1. Sheldom M. Ross. "Introductory Statistics,"?Section 7.4. Academic Press, 2017.

2. DePaul University. "NORMAL Distribution: Origin of the name."

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