bivariate distribution

(noun)

gives the probability that both of two random variables fall in a particular range or discrete set of values specified for that variable

Examples of bivariate distribution in the following topics:

  • Regression Toward the Mean: Estimation and Prediction

    • Regression towards the mean can be defined for any bivariate distribution with identical marginal distributions.
    • Not all such bivariate distributions show regression towards the mean under this definition.
    • However, all such bivariate distributions show regression towards the mean under the other definition.
    • Then, each student's score would be a realization of one of a set of independent and identically distributed random variables, with a mean of 50.

  • Graphing Bivariate Relationships

    • We can learn much more by displaying bivariate data in a graphical form that maintains the pairing of variables.
    • Measures of central tendency, variability, and spread summarize a single variable by providing important information about its distribution.
    • Each distribution is fairly skewed with a long right tail.
    • The presence of qualitative data leads to challenges in graphing bivariate relationships.
    • Compare the strengths and weaknesses of the various methods used to graph bivariate data.

  • Introduction to Bivariate Data

    • A dataset with two variables contains what is called bivariate data.
    • Measures of central tendency, variability, and spread summarize a single variable by providing important information about its distribution.
    • In this chapter we consider bivariate data, which for now consists of two quantitative variables for each individual.Our first interest is in summarizing such data in a way that is analogous to summarizing univariate (single variable) data.
    • Each distribution is fairly skewed with a long right tail.
    • We can learn much more by displaying the bivariate data in a graphical form that maintains the pairing.

  • Tukey Ladder of Powers

    • Many statistical methods such as t tests and the analysis of variance assume normal distributions.
    • Although these methods are relatively robust to violations of normality, transforming the distributions to reduce skew can markedly increase their power.
    • The demonstration in Figure 7 shows distributions of the data from the Stereograms case study as transformed with various values of λ.
    • Decreasing λ makes the distribution less positively skewed.
    • Distribution of data from the Stereogram case study for various values of λ

  • Statistical Graphics

    • Many familiar forms, including bivariate plots, statistical maps, bar charts, and coordinate paper were used in the 18th century.
    • • Multivariate distribution and correlation in the late 19th and 20th century.

  • Glossary

    • Bivariate data is data for which there are two variables for each observation.
    • Many empirical distributions are approximated well by mathematical distributions such as the normal distribution.
    • A distribution with long tails relative to a normal distribution is leptokurtic.
    • One of the most common continuous distributions, a normal distribution is sometimes referred to as a "bell-shaped distribution. " If μ is the distribution mean, and σ the standard deviation, then the height (ordinate) of the normal distribution is given by
    • A distribution with short tails relative to a normal distribution is platykurtic.

  • Introduction

    • The type of data described in the examples is bivariate data - "bi" for two variables.

  • Summary

    • Bivariate Data: Each data point has two values.

  • Lab 2: Regression (Textbook Cost)

    • Collect bivariate data (number of pages in a textbook, the cost of the textbook).

  • Practice: Linear Regression

    • The student will evaluate bivariate data and determine if a line is an appropriate fit to the data.