Discrete vs Continuous Variables (With Examples)

A variable in statistics is an unknown numerical quantity whose value cannot be known in advance but for whom various probabilities can be calculated.

For example, the value taken by the uppermost face of a die on being thrown is a variable but we know that there is a one out of six chance that a particular number appears on the uppermost face. Such variables are called random variables in statistics.

The variables that we encounter in probability are of two types: Discrete and Continuous

Discrete Variables

A variable is said to be a discrete variable if it takes either finite or countably infinite values. For example, the number appearing on the uppermost face of the die is a discrete variable because only finitely many values are possible (those values being 1,2,3,4,5 and 6).

Another example of a discrete variable is the number of people who are born per year because it can take countably infinite values. By countably infinite we mean that it is possible to “count” this quantity.

As opposed to a continuous variable, discrete variables take values in “steps” like 1,2,3,4,…… and hence it is possible to count them.

A Discrete variable has the following three properties:

1. Discrete Variables have finite or countably infinite values.
2. Discrete variables are counted, not measured.
3. Discrete variables can only take specific values that cannot be divided. 

Examples of Discrete Variables

1. Money in your bank

The amount of money in your bank is a discrete variable because it is countable. Even if you tried counting the money in everyone’s bank, it would still be discrete since the sum of all money is countably finite.

Money in the bank also fulfills the second and third conditions; it is something that is counted, not measured; and it cannot be divided into decimal values like 5.365.

2. Age in Years

Age in years would be a discrete variable since it can be counted. For example, one can be 17 years old. 

However, if one were to try to calculate age with exact precision, it could take uncountably infinite values. For example, your present age could be 21 years, 10 months, 4 days, 4 hours, 19 seconds, 56 milliseconds, 34 nanoseconds, 64 picoseconds…..and so on. In that case, age would not be a discrete variable but a continuous variable. 

Some further examples of discrete variables are:

1. The number on the uppermost face of the dice.
2. The number of printing errors per page on a book.
3. The number of customers arriving at a restaurant.

The probabilities associated with the discrete random variable are given by a probability mass function (pmf).

Some famous examples of discrete distributions are the binomial distribution, Poisson distribution, multinomial distribution, geometric distribution, etc.

Discrete Variables Practice Examples With Answers

Now, to ensure you have understood the concept of discrete variables, try to think if the following would be discrete variables or not:

1. Country of Birth
2. Weight in Kilograms 
3. Precise Weight in decimals
4. Number of books on the shelf in the library

Answers:

1. Discrete variable
2. Discrete variable
3. Not a Discrete variable
4. Discrete variable

Continuous Variables

A variable is said to be continuous if it takes uncountably infinite values, that is, it is not possible to “count” these quantities. These quantities do not occur in steps like 1,2,3,4…. and cannot usually be determined to accurate precision.

For example, height is a continuous variable because we can only determine the height of a person up to some decimal precision like 1.72m whereas the actual height might be something like 1.7222……so on.

A continuous variable has the following three properties:

1. It has uncountably infinite values, which means you cannot count it in a step-by-step manner.
2. Continuous variables are usually measured, not counted. Think temperature, speed of a vehicle, the height of a person, the weight of a person, time, etc.
3. Continuous variables can be split into smaller parts, taking up valid fractional or decimal values.

Examples of Continuous Variables

1. Age

Age is a continuous variable because it could take uncountably infinite values if we tried to count your age with exact precision. For example, your present age could be 31 years, 5 months, 8 days, 10 hours, 15 seconds, 10 milliseconds, 5 nanoseconds, 78 picoseconds…..and so on.

However, if you were to specifically define your age in terms of years, your age would have a countable and finite value. For example, your age in years would only be 31 years. In this case, age would not be a continuous variable, it would be a discrete variable.

2. Winning time in a race 

Winning time in a race is a continuous variable because the exact precision of the winning time has uncountably infinite values. For example, the winning time could be something like 50 seconds, 18 milliseconds, 9 nanoseconds, 52 picoseconds…..and so on.

However, if we were to specifically define the winning time using only up to two decimal places, then it would be countable and finite. For example, the winning time would be 50 seconds only. In this case, the winning time would be a continuous variable, not a discrete variable. 

Some more examples of continuous variables are:

1. Height of a person.
2. Weight of a person.
3. Time a customer waits in a bank queue.
4. The lifetime of a bulb or any electronic device.

The probabilities associated with the continuous random variable are given by a probability density function (pdf).

Some famous examples of continuous distributions are normal distribution, Cauchy distribution, Pareto distribution, Weibull distribution, etc.

Continuous Variables Practice Examples With Answers

To make sure you have developed a proper understanding of continuous variables, try to think if the following would be continuous variables or not:

1. Weight of a person
2. Weight of a person defined specifically in kilograms
3. Speed of bicycle.
4. Number of heads when flipping a coin

Answers:

1. Continuous variable
2. Not a continuous variable
3. Continuous variable
4. Not a continuous variable