# 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](https://www.investopedia.com/terms/b/bell-curve.asp)." 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 Takeaways

*   The normal distribution, also known as the Gaussian distribution, is a bell-shaped curve that is symmetric around the mean, indicating that most data points cluster near the mean.
*   It is characterized by two parameters: mean (average) and standard deviation, with crucial properties such as skewness and kurtosis that help describe the distribution's symmetry and tails.
*   The empirical rule states that for a normal distribution, approximately 68.2% of data falls within one standard deviation of the mean, 95.4% within two, and 99.7% within three, marking rare events as those falling beyond three standard deviations.
*   Despite its widespread use in finance, normal distribution assumptions can be misleading for asset prices, which often exhibit fat tails and skewed distributions, challenging forecast accuracy.
*   Normal distribution underpins many statistical theories, including the Central Limit Theorem, though reliance on it in finance must be tempered with awareness of its limitations given the complex nature of market behaviors.

![Normal Distribution](https://www.investopedia.com/thmb/6yrgTUN2KUGz1gJhmC3Kk_m0Kl4=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Normal_Distribution-8717e74027154b74a97567871ca741b0.jpg)

*— Investopedia / Lara Antal*

## Key Properties of Normal Distribution Explained

In technical stock market analysis, the normal distribution is commonly assumed. It has [two parameters](https://www.investopedia.com/articles/active-trading/092914/normal-distribution-table-explained.asp): mean and standard deviation. Here, [mean](https://www.investopedia.com/terms/m/mean.asp) (average), [median](https://www.investopedia.com/terms/m/median.asp) (midpoint), and [mode](https://www.investopedia.com/terms/m/mode.asp) (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](https://www.investopedia.com/terms/c/central_limit_theorem.asp) (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](https://www.investopedia.com/terms/s/symmetrical-distribution.asp). 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.

![Normal Distribution](https://www.investopedia.com/thmb/wHhPGldMI0COU2gr6FkEaeiYtCE=/941x0/filters:no_upscale():max_bytes(150000):strip_icc()/dotdash_final_Optimize_Your_Portfolio_Using_Normal_Distribution_23-0701d4f1047045a392ba624b68a8b8be.jpg)

*— Investopedia / Sabrina Jiang*

## Analyzing Skewness in Normal Distribution

Skewness measures the degree of symmetry of a distribution. The normal distribution is symmetric and has a [skewness](https://www.investopedia.com/terms/s/skewness.asp) 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.

![Skewness](https://www.investopedia.com/thmb/9Sh6pI1s-uDWQ97tZ_usQvOpMnI=/440x0/filters:no_upscale():max_bytes(150000):strip_icc()/skewness-5bfc2ac6c9e77c005876e4fd.png)

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## Exploring Kurtosis in Normal Distribution

[Kurtosis](https://www.investopedia.com/terms/k/kurtosis.asp) measures the thickness of the tail ends of a distribution to the tails of a distribution. The normal distribution has a [kurtosis](https://www.investopedia.com/terms/k/kurtosis.asp) 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](https://www.investopedia.com/terms/e/excesskurtosis.asp) is known in statistics as [leptokurtic](https://www.investopedia.com/terms/l/leptokurtic.asp), but is more colloquially known as "fat tails." The occurrence of fat tails in financial markets describes what is known as [tail risk](https://www.investopedia.com/terms/t/tailrisk.asp). Distributions with low kurtosis less than 3.0 ([platykurtic](https://www.investopedia.com/terms/p/platykurtic.asp)) exhibit tails that are generally less extreme ("skinnier") than the tails of the normal distribution.

![Kurtosis gives a numerical representation of the type of peak for a distribution.](https://www.investopedia.com/thmb/xLQ8Ru1bAACAHmfeGYmfKHNdteE=/1402x0/filters:no_upscale():max_bytes(150000):strip_icc()/kurtosis-5764c7f25f9b58346a2433c3.jpg)

*Kurtosis describes the different kinds of peaks that probability distributions can have.*

## 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

![Normal Distribution Formula](https://www.investopedia.com/thmb/5gtWqpcrakvHj0ky_vyQPvDI41Y=/1069x0/filters:no_upscale():max_bytes(150000):strip_icc()/Clipboard01-fdb217713438416cadafc48a1e4e5ee4.jpg)

*Normal Distribution Formula.*

  
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](https://www.investopedia.com/terms/p/price-action.asp). 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](https://www.investopedia.com/terms/c/capm.asp) 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).

![Heights](https://www.investopedia.com/thmb/DWJtVfvADdEsGYjRMnwp5LyIrY4=/864x0/filters:no_upscale():max_bytes(150000):strip_icc()/HtsEmpirical-1-d8d33ce1b8a64870a8cd0d67f1d8bf0d.png)

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## 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?"

Known technically as the Gaussian distribution, it was termed "normal" after 19th-century studies showed many natural events "normally deviated" around the mean. Sir Francis Galton popularized this as the "normal curve" in 1889.

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

Although normal distribution is a 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. Prices tend to follow more of a [log-normal distribution](https://www.investopedia.com/articles/investing/102014/lognormal-and-normal-distribution.asp), right-skewed and with fatter tails. Therefore, relying too heavily on a bell curve when making predictions 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.

## 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.

> **Article Sources**
>
> 1.  DePaul University. "[Normal Distribution: Origin of the Name](https://condor.depaul.edu/ntiourir/NormalOrigin.htm)."