continuous variable

Examples of continuous variable in the following topics:

• Introduction

  • Continuous random variables have many applications.
  • The field of reliability depends on a variety of continuous random variables.
  • This chapter gives an introduction to continuous random variables and the many continuous distributions.
  • NOTE: The values of discrete and continuous random variables can be ambiguous.
  • If X is the distance you drive to work, then you measure values of X and X is a continuous random variable.

• Types of Variables

  • Numeric variables may be further described as either continuous or discrete.
  • A continuous variable is a numeric variable.
  • Examples of continuous variables include height, time, age, and temperature.
  • Variables can be numeric or categorial, being further broken down in continuous and discrete, and nominal and ordinal variables.
  • Distinguish between quantitative and categorical, continuous and discrete, and ordinal and nominal variables.

• Recognizing and Using a Histogram

  • First introduced by Karl Pearson, it is an estimate of the probability distribution of a continuous variable.
  • The rectangles of a histogram are drawn so that they touch each other to indicate that the original variable is continuous.
  • As mentioned, a histogram is an estimate of the probability distribution of a continuous variable.
  • Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is, at most, countable.
  • If the distribution of $x$ is continuous, then $x$ is called a continuous random variable and, therefore, has a continuous probability distribution.

• Variables

  • In this case, the variable is "type of antidepressant. " When a variable is manipulated by an experimenter, it is called an independent variable.
  • An important distinction between variables is between qualitative variables and quantitative variables.
  • Qualitative variables are sometimes referred to as categorical variables.
  • Other variables such as "time to respond to a question" are continuous variables since the scale is continuous and not made up of discrete steps.
  • Of course, the practicalities of measurement preclude most measured variables from being truly continuous.

• Two Types of Random Variables

  • A random variable $x$, and its distribution, can be discrete or continuous.
  • As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.
  • Random variables can be classified as either discrete (that is, taking any of a specified list of exact values) or as continuous (taking any numerical value in an interval or collection of intervals).
  • Continuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function (CDF) that is absolutely continuous.
  • Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.

• Continuous Probability Distributions

  • A continuous probability distribution is a representation of a variable that can take a continuous range of values.
  • If the distribution of $X$ is continuous, then $X$ is called a continuous random variable.
  • Intuitively, a continuous random variable is the one which can take a continuous range of values—as opposed to a discrete distribution, in which the set of possible values for the random variable is at most countable.
  • While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3 and a half on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable.
  • In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value.

• Controlling for a Variable

  • Controlling for a variable is a method to reduce the effect of extraneous variations that may also affect the value of the dependent variable.
  • Histograms help us to visualize the distribution of data and estimate the probability distribution of a continuous variable.
  • Variables can be discrete (taking values from a finite or countable set), continuous (having a continuous distribution function), or neither.
  • For instance, temperature is a continuous variable, while the number of legs of an animal is a discrete variable.
  • In a scientific experiment measuring the effect of one or more independent variables on a dependent variable, controlling for a variable is a method of reducing the confounding effect of variations in a third variable that may also affect the value of the dependent variable.

• Types of variables

  • This variable seems to be a hybrid: it is a categorical variable but the levels have a natural ordering.
  • A variable with these properties is called an ordinal variable.
  • To simplify analyses, any ordinal variables in this book will be treated as categorical variables.
  • Classify each of the variables as continuous numerical, discrete numerical, or categorical.
  • Height varies continuously, so it is a continuous numerical variable.

• Using the Model for Estimation and Prediction

  • Standard multiple regression involves several independent variables predicting the dependent variable.
  • We would use standard multiple regression in which gender and weight would be the independent variables and height would be the dependent variable.
  • That is, it does not make sense to talk about the effect on height as gender increases or decreases, since gender is not a continuous variable.
  • As mentioned, the significance levels given for each independent variable indicate whether that particular independent variable is a significant predictor of the dependent variable, over and above the other independent variables.
  • This could happen because the covariance that the first independent variable shares with the dependent variable could overlap with the covariance that is shared between the second independent variable and the dependent variable.

• Two model selection strategies

  • Variables are eliminated one-at-a-time from the model until only variables with statistically significant p-values remain.
  • We continue in this way until removing variables does not increase $R^2_{adj}$.
  • It is common for p-values of one variable to change, due to collinearity, after eliminating a different variable.
  • Instead of eliminating variables one-at-a-time, we add variables one-at-a-time until we cannot find any variables that present strong evidence of their importance in the model.
  • Out of these four variables, the wheels variable had the smallest p-value.