Normal Distribution Curve

Normal Distribution Curve In statistics, the theoretical curve that shows how often an experiment will produce a particular result. The curve is symmetrical and bell shaped, showing that trials will usually give a result near the average, but will occasionally deviate by large amounts. The width of the “bell” indicates how much confidence one can have in the result of an experiment — the narrower the bell, the higher the confidence. This curve is also called the Gaussian curve, after the nineteenth-century German mathematician Karl Friedrich Gauss. Normal Distribution

Probability distribution. It has the following important characteristics: (1) the curve has a single peak; (2) it is bell-shaped; (3) the mean (average) lies at the center of the distribution, and the distribution is symmetrical around the mean; (4) the two tails of the distribution extend indefinitely and never touch the horizontal axis; (5) the shape of the distribution is determined by its Mean (µ) and Standard Deviation (s).

As with any continuous probability function, the area under the curve must equal 1, and the area between two values of X (say, a and b) represents the probability that X lies between a and b as illustrated on Figure 1. Further, since the normal is a symmetric distribution, it has the nice property that a known percentage of all possible values of X lie within ± a certain number of standard deviations of the mean Skewness Skewness is a parameter that describes asymmetry in a random variable’s probability distribution. Both probability density functions (PDFs) in Exhibit 1 have the same mean and standard deviation. The one on the left is positively skewed.The right tail is the longest; the mass of the distribution is concentrated on the left of the figure. The distribution is said to be right-skewed. The one on the right is negatively skewed. The left tail is the longest; the mass of the distribution is concentrated on the right of the figure. The distribution is said to be left-skewed.

Positive vs. Negative Skewness Exhibit 1

These graphs illustrate the notion of skewness. Both PDFs have the same expectation and variance. The one on the left is positively skewed. The one on the right is negatively skewed. The skewness of a random variable X is denoted

or skew(X). It is defined as:

where and are the mean and standard deviation of X. As one might expect, the formula takes on a positive value if X is positively skewed and a negative value if X is negatively skewed.

Kurtosis Kurtosis is a parameter that describes the shape of a random variable’s probability density function (PDF). Consider the two PDFs in Exhibit 1: Low Exhibit 1

vs.

High

Kurtosis

These graphs illustrate the notion of kurtosis. The PDF on the right has higher kurtosis than the PDF on the left. It is more peaked at the center, and it has fatter tails. Which would you say has the greater standard deviation? It is impossible to say. The PDF on the right is more peaked at the center, which might lead us to believe that it has a lower standard deviation. It has fatter tails, which might lead us to believe that it has a higher standard deviation. If the effect of the peakedness exactly offsets that of the fat tails, the two PDFs will have the same standard deviation. The different shapes of the two PDFs illustrate kurtosis. The PDF on the right has a greater kurtosis than the one on the left. The kurtosis of a random variable X is denoted

or kurt(X). It is defined as

where and are the mean and standard deviation of X. A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is said to be leptokurtic. If its kurtosis is less than 3, it is said to be platykurtic. Leptokurtosis is associated with PDFs that are simultaneously “peaked” and have “fat tails.” Platykurtosis is associated with PDFs that are simultaneously less peaked and have thinner tails. They are said to have "shoulders." In Exhibit 1, the PDF on the left is platykurtic. The one on the right is leptokurtic.

http://www.answers.com/normal+curves+?cat=biz-fin http://www.answers.com/skewness?cat=biz-fin http://www.riskglossary.com/link/kurtosis.htm