Pareto Distribution

(noun)

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena.

Related Terms

Examples of Pareto Distribution in the following topics:

  • Shapes of Sampling Distributions

    • The "shape of a distribution" refers to the shape of a probability distribution.
    • The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central; and where types of departure from this include:
    • A normal distribution is usually regarded as having short tails, while a Pareto distribution has long tails.
    • Even in the relatively simple case of a mounded distribution, the distribution may be skewed to the left or skewed to the right (with symmetric corresponding to no skew).
    • Sample distributions, when the sampling statistic is the mean, are generally expected to display a normal distribution.

  • Do It Yourself: Plotting Qualitative Frequency Distributions

    • Qualitative frequency distributions can be displayed in bar charts, Pareto charts, and pie charts.
    • Sometimes a relative frequency distribution is desired.
    • A special type of bar graph where the bars are drawn in decreasing order of relative frequency is called a Pareto chart .
    • This pie chart shows the frequency distribution of a bag of Skittles.
    • This graph shows the frequency distribution of a bag of Skittles.

  • The Gauss Model

    • The normal (Gaussian) distribution is a commonly used distribution that can be used to display the data in many real life scenarios.
    • In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution, defined by the formula:
    • A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
    • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
    • Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.

  • Exploratory Data Analysis (EDA)

    • His reasoning was that the median and quartiles, being functions of the empirical distribution, are defined for all distributions, unlike the mean and standard deviation.
    • Moreover, the quartiles and median are more robust to skewed or heavy-tailed distributions than traditional summaries (the mean and standard deviation).

  • t Distribution Table

    • T distribution table for 31-100, 150, 200, 300, 400, 500, and infinity.

  • Student Learning Outcomes

  • The Normal Distribution

    • Normal distributions are a family of distributions all having the same general shape.
    • The normal distribution is a continuous probability distribution, defined by the formula:
    • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
    • The simplest case of normal distribution, known as the Standard Normal Distribution, has expected value zero and variance one.
    • Many sampling distributions based on a large $N$ can be approximated by the normal distribution even though the population distribution itself is not normal.

  • Chi Square Distribution

    • Define the Chi Square distribution in terms of squared normal deviates
    • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
    • The area of a Chi Square distribution below 4 is the same as the area of a standard normal distribution below 2, since 4 is 22.
    • As the degrees of freedom increases, the Chi Square distribution approaches a normal distribution.
    • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.

  • The t-Distribution

    • Student's $t$-distribution (or simply the $t$-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
    • The $t$-distribution with $n − 1$ degrees of freedom is the sampling distribution of the $t$-value when the samples consist of independent identically distributed observations from a normally distributed population.
    • The $t$-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth.
    • Fisher, who called the distribution "Student's distribution" and referred to the value as $t$.
    • Note that the $t$-distribution becomes closer to the normal distribution as $\nu$ increases.

  • Common Discrete Probability Distribution Functions

    • Your instructor will let you know if he or she wishes to cover these distributions.
    • A probability distribution function is a pattern.
    • These distributions are tools to make solving probability problems easier.
    • Each distribution has its own special characteristics.
    • Learning the characteristics enables you to distinguish among the different distributions.