5 examples of discrete and continuous variables

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5 examples of discrete and continuous variables

Discrete and Continuous Random Variables: A variable is a quantity whose value changes. A discrete variable is a variable whose value is obtained by counting. Examples: number of students present

number of red marbles in a jar

number of heads when flipping three coins

students' grade level A continuous variable is a variable whose value is obtained by measuring. Examples: height of students in class

weight of students in class

time it takes to get to school

distance traveled between classes A random variable is a variable whose value is a numerical outcome of a random

phenomenon.

A random variable is denoted with a capital letter

The probability distribution of a random variable X tells what the possible values of X are and how probabilities are assigned to those values

A random variable can be discrete or continuous A discrete random variable X has a countable number of possible values.

Example: Let X represent the sum of two dice. Then the probability distribution of X is as follows: X 2 3 4 5 6 7 8 9 10 11 12 P(X) To graph the probability distribution of a discrete random variable, construct a probability histogram. A continuous random variable X takes all values in a given interval of numbers.

The probability distribution of a

continuous random variable is shown by a density curve.

The probability that X is between an interval of numbers is the area under the density curve between the interval endpoints

The probability that a continuous random variable X is exactly equal to a number is zero Means and Variances of Random Variables: The mean of a discrete

random variable, X, is its weighted average. Each value of X is weighted by its probability. To find the mean of X, multiply each value of X by its probability, then add all the products. The mean of a random variable X is called the expected value of X. Law of Large Numbers: As the number of observations increases, the mean of the observed values, ,

approaches the mean of the population, . The more variation in the outcomes, the more trials are needed to ensure that is close to . Rules for Means: If X is a random variable and a and b are fixed numbers, then If X and Y are random variables, then Example: Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math

score, Y. Suppose the average PSAT math score is 48. What is the average SAT math score?

Example: Let represent the average SAT math score. Let represent the average SAT verbal score. represents the average combined SAT score. Then is the average combined total SAT score. The Variance of a Discrete Random Variable: If X

is a discrete random variable with mean , then the variance of X is The standard deviation is the square root of the variance. Rules for Variances: If X is a random variable and a and b are fixed numbers, then If X and Y are independent random variables, then

Example: Suppose the equation Y = 20 + 100X converts a PSAT math score,

X, into an SAT math score, Y. Suppose the standard deviation for the PSAT math score is 1.5 points. What is the standard deviation for the SAT math score? Suppose the standard deviation for the SAT math score is 150 points, and the standard deviation for the SAT verbal score is 165 points. What is the standard deviation for the combined SAT

score? *** Because the SAT math score and SAT verbal score are not independent, the rule for adding variances does not apply! Variable refers to the quantity that changes its value, which can be measured. It is of two types, i.e. discrete or continuous variable. The former refers to the one that has a certain number of values, while the latter implies

the one that can take any value between a given range. Data can be understood as the quantitative information about a specific characteristic. The characteristic can be qualitative or quantitative, but for the purpose of statistical analysis, the qualitative characteristic is transformed into quantitative one, by providing numerical data of that

characteristic. So, the quantitative characteristic is known as a variable. Here in this article, we are going to talk about the discrete and continuous variable. Content: Discrete Variable Vs Continuous Variable Comparison Chart Definition Key Differences Examples Conclusion Comparison Chart Basis for ComparisonDiscrete VariableContinuous

Variable MeaningDiscrete variable refers to the variable that assumes a finite number of isolated values.Continuous variable alludes to the a variable which assumes infinite number of different values. Range of specified numberCompleteIncomplete ValuesValues are obtained by counting.Values are obtained by measuring. ClassificationNon-

overlappingOverlapping AssumesDistinct or separate values.Any value between the two values. Represented byIsolated pointsConnected points Definition of Discrete Variable A discrete variable is a type of statistical variable that can assume only fixed number of distinct values and lacks an inherent order. Also known as a categorical variable,

because it has separate, invisible categories. However no values can exist in-between two categories, i.e. it does not attain all the values within the limits of the variable. So, the number of permitted values that it can suppose is either finite or countably infinite. Hence if you are able to count the set of items, then the variable is said to be discrete.

Definition of Continuous Variable Continuous variable, as the name suggest is a random variable that assumes all the possible values in a continuum. Simply put, it can take any value within the given range. So, if a variable can take an infinite and uncountable set of values, then the variable is referred as a continuous variable. A continuous variable is

one that is defined over an interval of values, meaning that it can suppose any values in between the minimum and maximum value. It can be understood as the function for the interval and for each function, the range for the variable may vary. The difference between discrete and continuous variable can be drawn clearly on the following grounds:

The statistical variable that assumes a finite set of data and a countable number of values, then it is called as a discrete variable. As against this, the quantitative variable which takes on an infinite set of data and a uncountable number of values is known as a continuous variable. For non-overlapping or otherwise known as mutually inclusive

classification, wherein the both the class limit are included, is applicable for the discrete variable. On the contrary, for overlapping or say mutually exclusive classification, wherein the upper class-limit is excluded, is applicable for a continuous variable. In discrete variable, the range of specified number is complete, which is not in the case of a

continuous variable. Discrete variables are the variables, wherein the values can be obtained by counting. On the other hand, Continuous variables are the random variables that measure something. Discrete variable assumes independent values whereas continuous variable assumes any value in a given range or continuum. A discrete variable can be

graphically represented by isolated points. Unlike, a continuous variable which can be indicated on the graph with the help of connected points. Examples Discrete Variable Number of printing mistakes in a book. Number of road accidents in New Delhi. Number of siblings of an individual. Continuous Variable Height of a person Age of a person Profit

earned by the company. Conclusion By and large, both discrete and continuous variable can be qualitative and quantitative. However, these two statistical terms are diametrically opposite to one another in the sense that the discrete variable is the variable with the well-defined number of permitted values whereas a continuous variable is a variable

that can contain all the possible values between two numbers. The reason why we often class variables into different types is because not all statistical analyses can be performed on all variable types. For instance, it is impossible to compute the mean of the variable "hair color" as you cannot sum brown and blond hair. On the other hand, finding the

mode of a continuous variable does not really make any sense because most of the time there will not be two exact same values, so there will be no mode. And even in the case there is a mode, there will be very few observations with this value. As an example, try finding the mode of the height of the students in your class. If you are lucky, a couple of

students will have the same size. However, most of the time, every student will have a different size (especially if heights have been measured in millimeters) and thus there will be no mode. To see what kind of analysis is possible on each type of variable, see more details in the articles "Descriptive statistics by hand" and "Descriptive statistics in R".

Similarly, some statistical tests can only be performed on certain type of variables. For example, a correlation can only be computed on quantitative variables, while a Chi-square test of independence is done with qualitative variables, and a Student t-test or ANOVA requires a mix of quantitative and qualitative variables. Last but not least, in datasets

it is very often the case that numbers are used for qualitative variables. For instance, a researcher may assign the number "1" to women and the number "2" to men (or "0" to the answer "No" and "1" to the answer "Yes"). Despite the numerical classification, the variable gender is still a qualitative variable and not a discrete variable as it may look.

The numerical classification is only used to facilitate data collection and data management. It is indeed easier to write the number "1" or "2" instead of "women" or "men", and thus less prone to encoding errors. If you face this kind of setup, do not forget to transform your variable into the right type before performing any statistical analyses. Usually,

a basic descriptive analysis (and knowledge about the variables which have been measured) prior to the main statistical analyses is enough to check that all variable types are correct. Thanks for reading. I hope this article helped you to understand the different types of variable. If you would like to learn more about the different data types in R, read

the article "Data types in R". As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion.

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