Normal Distribution
The normal distribution is a continuous probability distribution that, when graphed as a probability density, takes the form of the so-called bell-shaped curve. The bell shape results from the fact that, while the range of possible outcomes is infinite (negative infinity to positive infinity), most of the potential outcomes tend to be clustered relatively close to the distribution's mean value. Just how close they are clustered is given by the standard deviation. In other words, a normal distribution is described completely by two parameters: its mean (μ) and its standard deviation (σ).

Here are other defining characteristics of the normal distribution: it is symmetric, meaning the mean value divides the distribution in half and one side is the exact mirror image of the other -that is, skewness = 0. Symmetry also requires that mean = median = mode. Its kurtosis (measure of peakedness) is 3 and its excess kurtosis (kurtosis - 3) equals 0. Also, if given 2 or more normally distributed random variables, the linear combination must also be normally distributed.

While any normal distribution will share these defining characteristics, the mean and standard deviation will be unique to the random variable, and these differences will affect the shape of the distribution. On the following page are two normal distributions, each with the same mean, but the distribution with the dotted line has a higher standard deviation.

Univariate vs. Multivariate Distributions
A univariate distribution specifies probabilities for a single random variable, while a multivariate distribution combines the outcomes of a group of random variables and summarizes probabilities for the group. For example, a stock will have a distribution of possible return outcomes; those outcomes when summarized would be in a univariable distribution. A portfolio of 20 stocks could have return outcomes described in terms of 20 separate univariate distributions, or as one multivariate distribution.

Earlier we indicated that a normal distribution is completely described by two parameters: its mean and standard deviation. This statement is true of a univariate distribution. For models of multivariate returns, the mean and standard deviation of each variable do not completely describe the multivariate set. A third parameter is required, the correlation, or co-movement, between each pair of variables in the set. For example, if a multivariate return distribution was being assembled for a portfolio of stocks, and a number of pairs were found to be inversely related (i.e. one increases at the same time the other decreases), then we must consider the overall effect on portfolio variance. For a group of assets that are not completely positively related, there is the opportunity to reduce overall risk (variance) as a result of the interrelationships.

For a portfolio distribution with n stocks, the multivariate distribution is completely described by the n mean returns, the n standard deviations and the n*(n - 1)/2 correlations. For a 20-stock portfolio, that's 20 lists of returns, 20 lists of variances of return and 20*19/2, or 190 correlations.

Confidence Intervals

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

    The Uses And Limits Of Volatility

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

    Multivariate Models: The Monte Carlo Analysis

    This decision-making tool integrates the idea that every decision has an impact on overall risk.
  5. Investing

    Lognormal and Normal Distribution

    When and why do you use lognormal distribution or normal distribution for analyzing securities? Lognormal for stocks, normal for portfolio returns.
  6. Investing

    Stock Market Risk: Wagging The Tails

    The bell curve is an excellent way to evaluate stock market risk over the long term.
  7. Investing

    Scenario Analysis Provides Glimpse Of Portfolio Potential

    This statistical method estimates how far a stock might fall in a worst-case scenario.
  8. Financial Advisor

    How to Save Clients from RMD Aggregation Mistakes

    Advisors can help clients avoid required minimum distribution mistakes in their retirement plans.
Frequently Asked Questions
  1. What is the history of the S&P 500?

    Discover the history of the S&P 500, which sophisticated market participants consider to be the best index to understand ...
  2. What is the formula for calculating weighted average cost of capital (WACC) in Excel?

    Learn about the weighted average cost of capital (WACC) formula and how it is used to estimate the average cost of raising ...
  3. Where do most fund managers get their market information?

    Many fund managers, whether they manage a mutual fund, trust fund, pension or hedge fund, have access to resources that the ...
  4. What's the difference between short-term investments and marketable securities?

    Understand the difference between short-term investments and marketable equity securities, and learn the importance of short-term ...
Trading Center