Normal Distribution

Loading the player...

What is the 'Normal Distribution'

The normal distribution, also known as the Gaussian or standard normal distribution, is the probability distribution that plots all of its values in a symmetrical fashion, and most of the results are situated around the probability's mean. Values are equally likely to plot either above or below the mean. Grouping takes place at values close to the mean and then tails off symmetrically away from the mean.

BREAKING DOWN 'Normal Distribution'

The normal distribution is the most common type of distribution and is often found in stock market analysis. Given enough observations within a sample size, it is reasonable to make the assumption that returns follow a normally distributed pattern, but this assumption can be disproved.

As with any distribution, the distribution's mean, skewness and kurtosis coefficients should be calculated to determine the type of distribution. The standard normal distribution has two parameters: the mean and the standard deviation. In the Gaussian distribution, the mean, or mu, is equal to zero, while the standard deviation, or sigma, is equal to one. Under normal circumstances, independent identically distributed random variables, or outcomes, are said to converge to a standard normal distribution under the central limit theorem.

Skewness and Kurtosis

The skewness measures the symmetry of a distribution. The standard normal distribution has a skewness of zero, and therefore, it is said to be symmetric. If the distribution of a data set has a skewness less than zero, the data is skewed to the left. Conversely, data that has a positive skewness is said to be skewed to the right. For example, asset prices follow a lognormal distribution, which is skewed to the right because asset prices are non-negative.

The kurtosis measures the tail ends of a distribution and whether the distribution of a data set has skinny tails or fat tails in relation to the normal distribution. The standard normal distribution has a kurtosis of three, which indicates data that follow a Gaussian distribution have neither fat or thin tails. Therefore, if observed data have a kurtosis greater than three, it is said to have heavy tails when compared to the normal distribution. If the data have a kurtosis less than three, it is said to have thin tails.

For example, stock market returns are said to follow a normal distribution in theory. However, in reality, asset returns tend to have fat tails. Observed asset returns have typically had moves greater than three standard deviations beyond the mean more than expected under the assumptions of the normal distribution.

RELATED TERMS
  1. Tail Risk

    A form of portfolio risk that arises when the possibility that ...
  2. Kurtosis

    A statistical measure used to describe the distribution of observed ...
  3. Probability Distribution

    A statistical function that describes all the possible values ...
  4. Mesokurtic

    A term used in a statistical context where the kurtosis of a ...
  5. Symmetrical Distribution

    A situation in which the values of variables occur at regular ...
  6. Leptokurtic

    A statistical distribution where the points along the X-axis ...
Related Articles
  1. Investing

    What a Normal Distribution Means

    Normal distribution describes a symmetrical data distribution, where most of the results lie near the mean.
  2. Trading

    Trading With Gaussian Models Of Statistics

    The entire study of statistics originated from Gauss and allowed us to understand markets, prices and probabilities, among other applications.
  3. Investing

    Find The Right Fit With Probability Distributions

    Discover a few of the most popular probability distributions and how to calculate them.
  4. Trading

    What's Skewness?

    Skewness describes how a data distribution leans.
  5. ETFs & Mutual Funds

    Stock Market Risk: Wagging The Tails

    The bell curve is an excellent way to evaluate stock market risk over the long term.
  6. Managing Wealth

    Using Normal Distribution Formula To Optimize Your Portfolio

    Normal or bell curve distribution can be used in portfolio theory to help portfolio managers maximize return and minimize risk.
  7. Managing Wealth

    The Uses And Limits Of Volatility

    Check out how the assumptions of theoretical risk models compare to actual market performance.
  8. Investing

    Quantitative Analysis Of Hedge Funds

    Hedge fund analysis requires more than just the metrics used to analyze mutual funds.
  9. Markets

    Fat Tail Risk Makes Global Warming Scarier

    The cost of global warming does not take into account climate change-related catastrophes. Here's where fat-tail distributions come in.
  10. Trading

    A Simplified Approach To Calculating Volatility

    Though most investors use standard deviation to determine volatility, there's an easier and more accurate way of doing it.
RELATED FAQS
  1. Can I elect to NOT have income tax withheld from an IRA (NOT ROTH) distribution before ...

  2. Is my non-qualified Roth IRA distribution subject to taxes or early distribution ...

    The ordering rules must be applied to determine whether the distribution is subject to income taxes and/or the early distribution ... Read Answer >>
  3. Why are shareholder distributions eliminated and not added to the net income when ...

    When considering selling an S Corp. I am being told that shareholder distributions cannot be added back to determine its ... Read Answer >>
  4. Can I replace the $7,000 in Jan 2016 and avoid paying taxes on the $7,000 excess ...

    I closed an IRA account in Jan 2015 and received $9,000. I forgot that and withdrew my required $7,000 minimum distribution ... Read Answer >>
  5. What does Value at Risk (VaR) say about the "tail" of the loss distribution?

    Learn about value at risk and conditional value at risk and how both models interpret the tail ends of an investment portfolio's ... Read Answer >>
  6. If an early distribution has been ordered by the courts to pay back restitution, ...

    I went through the Federal Courts. I've spoken to my tax preparer, went to the local IRS office and no one has an answer ... Read Answer >>
Hot Definitions
  1. Duration

    A measure of the sensitivity of the price (the value of principal) of a fixed-income investment to a change in interest rates. ...
  2. Dove

    An economic policy advisor who promotes monetary policies that involve the maintenance of low interest rates, believing that ...
  3. Cyclical Stock

    An equity security whose price is affected by ups and downs in the overall economy. Cyclical stocks typically relate to companies ...
  4. Front Running

    The unethical practice of a broker trading an equity based on information from the analyst department before his or her clients ...
  5. After-Hours Trading - AHT

    Trading after regular trading hours on the major exchanges. The increasing popularity of electronic communication networks ...
  6. Omnibus Account

    An account between two futures merchants (brokers). It involves the transaction of individual accounts which are combined ...
Trading Center