Normal distributions are a family of distributions that have the same general shape.

• They are symmetrical with scores more concentrated in the middle than in the tails.
• Normal distributions are sometimes described as bell-shaped with a single peak at the exact center of the distribution.
• The tails of the normal curve extends indefinitely in both directions. That is, possible outcomes of a normal distribution lie between - ∞ and + ∞.
• Normal distributions may differ in how spread-out they are.

The graph looks like this:

The key properties are:

• The normal distribution is completely described by two parameters: the mean (μ) and the standard deviation (σ).
• The normal distribution is symmetrical: it has a skewness of 0, a kurtosis (it measures the peakedness of a distribution) of 3, and an excess kurtosis (which equals kurtosis less 3) of 0. As a consequence, the mean, median, and mode are all equal for a normal random variable.
• A linear combination of two or more normal random variables is also normally distributed.

One reason the normal distribution is important is that many psychological, educational, and financial variables are distributed approximately normally. Measures of reading ability, introversion, job satisfaction, and memory are among the many psychological variables approximately normally distributed. Although the distributions are only approximately normal, they are usually quite close.

A second reason is that it is easy for mathematical statisticians to work with. Many kinds of statistical tests can be derived for normal distributions. Almost all statistical tests discussed in the textbook assume normal distributions. Fortunately, these tests work very well even if the distribution is only approximately normally distributed. Some tests work well even with very wide deviations from normality.

Finally, if the mean and standard deviation of a normal distribution are known, it is easy to convert back and forth from raw scores to percentiles.

For example, normal distribution is an approximate model for asset returns. The price of any asset can only drop to 0. Therefore, the lowest return on an asset is -100% (i.e., all investment in the asset is lost). Since the normal distribution extends to negative infinity without limit, it is not an accurate model for asset returns. However, for the normal distribution, the probability of outcomes below -100% is very small. Therefore, the normal distribution can be considered an approximate model for returns. However, the normal distribution tends to underestimate the probability of extreme returns.

Confidence Intervals for a Normally Distributed Random Variable.

Analysts can use the sample mean to estimate the population mean, and the sample standard deviation to estimate the population standard deviation. The sample mean and sample standard deviation are point estimates.

Probability statements about a random variable are often framed using confidence intervals built around point estimates. In investment work, confidence intervals for a normal random variable in relation to its estimated mean are often used.

Confidence intervals use point estimates to make probability statements about the dispersion of the outcomes of a normal distribution. A confidence interval specifies the percentage of all observations that fall in a particular interval.

The exact confidence intervals for a normal random variable X:

• 90% confidence interval for X is: x-bar - 1.645σ to x-bar + 1.645σ: this means that 10% of the observations fall outside the 90% confidence interval, with 5% on each side.

• 95% confidence interval for X is: x-bar - 1.96σ to x-bar + 1.96 σ: this means that 5% of the observations fall outside the 95% confidence interval, with 2.5% on each side.

• 99% confidence interval for X is: x-bar - 2.58 σ to x-bar + 2.58 σ: this means that 1% of the observations fall outside the 99% confidence interval, with 0.5% on each side.

Hint: memorize these numbers (1.645, 1.96 and 2.58) to quickly solve relevant problems. For details about confidence intervals, refer to Reading 5 - Sampling and Estimation.

The Univariate and Multivariate Distributions

To this point, the focus has been on distributions that involve only one variable, such as the binomial, uniform, and normal distributions. A univariate distribution describes a single random variable. For example, suppose that you would like to model the distribution of the return on an asset. Such a distribution is a univariate distribution.

A multivariate distribution specifies the probabilities for a group of related random variables. It is used to describe the probabilities of a group of continuous random variables if all of the individual variables follow a normal distribution.

Each individual normal random variable would have its own mean and its own standard deviation, and hence its own variance. When you are dealing with two or more random variables in tandem, the strength of the relationship between (or among) the variables assumes huge importance. You will recall that the strength of the relationship between two random variables is known as the correlation.

When there is a group of assets, the distribution of returns on each asset can either be modeled individually or on the assets as a group. A multivariate normal distribution for the returns on n stocks is completely defined by three lists of parameters:

• The list of the mean returns on the individual securities (n means in total).
• The list of the securities' variances of return (n variances in total).
• The list of all the distinct pairwise return correlations (n(n-1)/2 distinct correlations in total).

The higher the correlation values, the higher the variance of the overall portfolio. In general, it is better to build a portfolio of stocks whose prices are not strongly correlated with each other, as this lowers the variance of the overall portfolio.

It is the correlation values that distinguish a multivariate normal distribution from a univariate normal distribution. Consider a portfolio consisting of 2 assets (n = 2). The multivariate normal distribution can be defined with 2 means, 2 variances, and 2 x (2-1)/2 = 1 correlation. If an analyst has a portfolio of 100 securities, the multivariate normal distribution can be defined with 100 means, 100 variances, and 100 x (100 - 1)/2 = 4950 correlations. Portfolio return is a weighted average of the returns on the 100 securities. A weighted average is a linear combination. Thus, portfolio return is normally distributed if the individual security returns are (joint) normally distributed. In order to specify the normal distribution for portfolio return, analysts need means, variances, and the distinct pairwise correlations of the component securities.